Question

Prove a seguinte igualdade

$sec ^{2} x \times cossec^{2}x = sec^{2} x + cossec^{2}$
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Answer 1

$\rightarrow~sec^2x~=~\frac{1}{cos^2x}\\\\\rightarrow~cossec^2x~=~\frac{1}{sin^2x}\\\\\\\\sec^2x~.~cossec^2x~=~sec^2x+cossec^2x\\\\\\\frac{1}{cos^2x}~.~\frac{1}{sin^2x}~=~\frac{1}{cos^2x}~+~\frac{1}{sin^2x}\\\\\\\frac{1}{cos^2x.sin^2x}~=~\frac{1~.~sin^2x~+~1~.~cos^2x}{cos^2x.sin^2x}\\\\\\\frac{1}{cos^2x.sin^2x}~=~\frac{sin^2x+cos^2x}{cos^2x.sin^2x}\\\\\\Aplicando~a~identidade~trigonometrica~sin^2x+cos^2x~=~1:\\\\\\\boxed{\frac{1}{cos^2x.sin^2x}~=~\frac{1}{cos^2x.sin^2x}}$