Question

(1+cotgx)² + (1-cotgx)² = 2.cossec²x

Alguém pode me ajudar com a identidade disso aí?

Answer 1

Resolvendo:

$\mathsf{\left(1+cotg\left(x\right)\right)^2+\left(1-cotg\left(x\right)\right)^2=2cossec^2\left(x\right)}\\\\\\\mathsf{1^2+2cotg\left(x\right)+cotg^2\left(x\right)+1^2-2cotg\left(x\right)+cotg^2\left(x\right)=2cossec^2\left(x\right)}\\\\\\\mathsf{2+2cotg^2\left(x\right)=2cossec^2\left(x\right)}\\\\\\\mathsf{2\cdot \left(1+cotg^2\left(x\right)\right)=2cossec^2\left(x\right)}\\\\\\\mathsf{1+cotg^2\left(x\right)=cossec^2\left(x\right)}$

$\\\\\\\mathsf{1+\dfrac{cos^2\left(x\right)}{sen^2\left(x\right)}=\dfrac{1}{sen^2\left(x\right)}}\\\\\\\mathsf{\dfrac{sen^2\left(x\right)}{sen^2\left(x\right)}+\dfrac{cos^2\left(x\right)}{sen^2\left(x\right)}=\dfrac{1}{sen^2\left(x\right)}}\\\\\\\mathsf{\dfrac{sen^2\left(x\right)+cos^2\left(x\right)}{sen^2\left(x\right)}=\dfrac{1}{sen^2\left(x\right)}}\\\\\\\mathsf{sen^2\left(x\right)+cos^2\left(x\right)=1}\\\\\\\mathsf{1=1}$

Logo, a igualdade é verdadeira.