Mathematics: Prove the identity csc^2xsec^2x=sec^2x+csc^2x

Prove the identity csc^2xsec^2x=sec^2x+csc^2x

Ответ:

lyssamichelle

14.03.2022

$\bf csc(\theta)=\cfrac{1}{sin(\theta)}
\qquad 
% secant
 sec(\theta)=\cfrac{1}{cos(\theta)}
\\\\\\
and\qquad sin^2(\theta)+cos^2(\theta)=1
\\\\
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csc^2(x)sec^2(x)=sec^2(x)+csc^2(x)\\\\
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sec^2(x)+csc^2(x)\implies \cfrac{1}{cos^2(x)}+\cfrac{1}{sin^2(x)}\implies \cfrac{sin^2(x)+cos^2(x)}{cos^2(x)sin^2(x)}
\\\\\\
\cfrac{1}{cos^2(x)sin^2(x)}\implies \cfrac{1}{cos^2(x)}\cdot \cfrac{1}{sin^2(x)}\implies sec^2(x)csc^2(x)$

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Prove the identity csc^2xsec^2x=sec^2x+csc^2x

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