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Popular Trigonometria >

provar 2sec(2x)=tan(pi/4+x)+tan(pi/4-x)

Solução

provar

2 sec (2 x) = tan (\frac{π}{4} + x) + tan (\frac{π}{4} - x)

Solução

Verdadeiro

Passos da solução

2 sec (2 x) = tan (\frac{π}{4} + x) + tan (\frac{π}{4} - x)

Manipular o lado esquerdo

tan (\frac{π}{4} + x) + tan (\frac{π}{4} - x)

Reeecreva usando identidades trigonométricas

tan (\frac{π}{4} + x)

Utilizar a Identidade Básica da Trigonometria:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( \frac{π}{4} + x )}{cos ( \frac{π}{4} + x )}

Use a identidade de soma de ângulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( \frac{π}{4} ) cos ( x ) + cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} + x )}

Use a identidade de soma de ângulos:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( \frac{π}{4} ) cos ( x ) + cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )}

Simplificar

\frac{sin ( \frac{π}{4} ) cos ( x ) + cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )} : \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( \frac{π}{4} ) cos ( x ) + cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )}

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + cos (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )}

cos (\frac{π}{4}) cos (x) - sin (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (\frac{π}{4}) cos (x) - sin (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - sin (\frac{π}{4}) sin (x)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}

Multiplicar

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

Multiplicar

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiplicar

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiplicar

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Combinar as frações usando o mínimo múltiplo comum:

\frac{2 cos ( x ) - 2 sin ( x )}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ( x ) - 2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

Combinar as frações usando o mínimo múltiplo comum:

\frac{2 cos ( x ) + 2 sin ( x )}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ( x ) + 2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

Dividir frações:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( 2 cos ( x ) + 2 sin ( x ) ) \cdot 2}{2 ( 2 cos ( x ) - 2 sin ( x ) )}

Eliminar o fator comum:

2

= \frac{2 cos ( x ) + 2 sin ( x )}{2 cos ( x ) - 2 sin ( x )}

Fatorar o termo comum

2

= \frac{2 ( cos ( x ) + sin ( x ) )}{2 cos ( x ) - 2 sin ( x )}

Fatorar o termo comum

2

= \frac{2 ( cos ( x ) + sin ( x ) )}{2 ( cos ( x ) - sin ( x ) )}

Eliminar o fator comum:

2

= \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )}

= \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )}

= \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + tan (\frac{π}{4} - x)

Reeecreva usando identidades trigonométricas

tan (\frac{π}{4} - x)

Utilizar a Identidade Básica da Trigonometria:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( \frac{π}{4} - x )}{cos ( \frac{π}{4} - x )}

Use a identidade de diferença de ângulos:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} - x )}

Use a identidade de diferença de ângulos:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

Simplificar

\frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )} : \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

\frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - cos (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{2}{2}

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (\frac{π}{4}) sin (x)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}

Multiplicar

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}

Multiplicar

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiplicar

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

\frac{2}{2} cos (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiplicar

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

\frac{2}{2} sin (x)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Combinar as frações usando o mínimo múltiplo comum:

\frac{2 cos ( x ) + 2 sin ( x )}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ( x ) + 2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

Combinar as frações usando o mínimo múltiplo comum:

\frac{2 cos ( x ) - 2 sin ( x )}{2}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ( x ) - 2 sin ( x )}{2}

= \frac{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

Dividir frações:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( 2 cos ( x ) - 2 sin ( x ) ) \cdot 2}{2 ( 2 cos ( x ) + 2 sin ( x ) )}

Eliminar o fator comum:

2

= \frac{2 cos ( x ) - 2 sin ( x )}{2 cos ( x ) + 2 sin ( x )}

Fatorar o termo comum

2

= \frac{2 ( cos ( x ) - sin ( x ) )}{2 cos ( x ) + 2 sin ( x )}

Fatorar o termo comum

2

= \frac{2 ( cos ( x ) - sin ( x ) )}{2 ( cos ( x ) + sin ( x ) )}

Eliminar o fator comum:

2

= \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

= \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

= \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

Simplificar

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} : \frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

Mínimo múltiplo comum de

cos (x) - sin (x), cos (x) + sin (x) : (cos (x) - sin (x)) (cos (x) + sin (x))

cos (x) - sin (x), cos (x) + sin (x)

Mínimo múltiplo comum (MMC)

Calcular uma expressão que seja composta por fatores que estejam presentes tanto em

cos (x) - sin (x)

quanto em

cos (x) + sin (x)

= (cos (x) - sin (x)) (cos (x) + sin (x))

Reescrever as frações baseando-se no mínimo múltiplo comum

Multiplicar cada numerador pelo mesmo valor necessário para multiplicar seu denominador correspondente para convertê-lo no mínimo múltiplo comum

Para

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} :

multiplique o numerador e o denominador por

cos (x) + sin (x)

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} = \frac{( cos ( x ) + sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = \frac{( cos ( x ) + sin ( x ) ) ^{2}}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

Para

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} :

multiplique o numerador e o denominador por

cos (x) - sin (x)

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} = \frac{( cos ( x ) - sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = \frac{( cos ( x ) - sin ( x ) ) ^{2}}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

= \frac{( cos ( x ) + sin ( x ) ) ^{2}}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} + \frac{( cos ( x ) - sin ( x ) ) ^{2}}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

Já que os denominadores são iguais, combinar as frações:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( cos ( x ) + sin ( x ) ) ^{2} + ( cos ( x ) - sin ( x ) ) ^{2}}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

Expandir

(cos (x) + sin (x))^{2} + (cos (x) - sin (x))^{2} : 2 cos^{2} (x) + 2 sin^{2} (x)

(cos (x) + sin (x))^{2} + (cos (x) - sin (x))^{2}

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

Aplique a fórmula do quadrado perfeito:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

= cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x) + (cos (x) - sin (x))^{2}

(cos (x) - sin (x))^{2} : cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x)

Aplique a fórmula do quadrado perfeito:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x)

= cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x) + cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x)

Simplificar

cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x) + cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x) : 2 cos^{2} (x) + 2 sin^{2} (x)

cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x) + cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x)

Somar elementos similares:

2 cos (x) sin (x) - 2 cos (x) sin (x) = 0

= cos^{2} (x) + sin^{2} (x) + cos^{2} (x) + sin^{2} (x)

Somar elementos similares:

cos^{2} (x) + cos^{2} (x) = 2 cos^{2} (x)

= 2 cos^{2} (x) + sin^{2} (x) + sin^{2} (x)

Somar elementos similares:

sin^{2} (x) + sin^{2} (x) = 2 sin^{2} (x)

= 2 cos^{2} (x) + 2 sin^{2} (x)

= 2 cos^{2} (x) + 2 sin^{2} (x)

= \frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

= \frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

Fatorar

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} : \frac{2 ( cos ^{2} ( x ) + sin ^{2} ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Fatorar

2 cos^{2} (x) + 2 sin^{2} (x) : 2 (cos^{2} (x) + sin^{2} (x))

2 cos^{2} (x) + 2 sin^{2} (x)

Fatorar o termo comum

2

= 2 (cos^{2} (x) + sin^{2} (x))

= \frac{2 ( cos ^{2} ( x ) + sin ^{2} ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

= \frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) \cdot 2}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Reeecreva usando identidades trigonométricas

\frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) \cdot 2}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Utilizar a identidade trigonométrica pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{1 \cdot 2}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Simplificar

= \frac{2}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Expandir

(cos (x) + sin (x)) (cos (x) - sin (x)) : cos^{2} (x) - sin^{2} (x)

(cos (x) + sin (x)) (cos (x) - sin (x))

Aplicar a regra da diferença de quadrados:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) - sin^{2} (x)

= \frac{2}{cos ^{2} ( x ) - sin ^{2} ( x )}

Utilizar a identidade trigonométrica do arco duplo:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= \frac{2}{cos ( 2 x )}

= \frac{2}{cos ( 2 x )}

Reeecreva usando identidades trigonométricas

Utilizar a Identidade Básica da Trigonometria:

cos (x) = \frac{1}{sec ( x )}

\frac{2}{\frac{1}{s e c ( 2 x )}}

Simplificar

\frac{2}{\frac{1}{s e c ( 2 x )}}

Aplicar as propriedades das frações:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{2 sec ( 2 x )}{1}

Aplicar a regra

\frac{a}{1} = a

= 2 sec (2 x)

2 sec (2 x)

2 sec (2 x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

Exemplos populares

provar sin(x)-csc(x)=-(cos(x))(cot(x))

p ro v e sin (x) - csc (x) = - (cos (x)) (cot (x))

provar cos(θ)=sin(3θ+62)

p ro v e cos (θ) = sin (3 θ + 62)

provar sin(pi*x)+sin(pi*(10-x))=0

p ro v e sin (π \cdot x) + sin (π \cdot (10 - x)) = 0

provar cot(2θ)= 1/2 (tan(θ)-cot(θ))

p ro v e cot (2 θ) = \frac{1}{2} (tan (θ) - cot (θ))

provar sin(4x)=(2)(sin(2x))(cos(2x))

p ro v e sin (4 x) = (2) (sin (2 x)) (cos (2 x))