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Question Number 126610 by bramlexs22 last updated on 22/Dec/20

$$Given\{\begin{array}{l}\cos x+\sin x=p\\ \sec x+\csc x=q\end{array}$$$$p\ne q.Find\frac{\cos {}^{2}x}{2-2\sin {}^{2}x-\mathrm{cs}{c}^{2}x}.$$

Answered by liberty last updated on 22/Dec/20

$$q=\left(\frac{\cos x+\sin x}{\cos x\sin x}\right)\Rightarrow q^2=\frac{p^2}{\cos {}^{2}x\sin {}^{2}x}$$$$\cos {}^{2}x\sin {}^{2}x=\frac{p^2}{q^2}...\left(i\right)$$$$\Rightarrow \frac{\cos {}^{2}x}{2\cos {}^{2}x-\frac{1}{\sin {}^{2}x}}=\frac{\cos {}^{2}x\sin {}^{2}x}{2\cos {}^{2}x\sin {}^{2}x-1}$$$$=\frac{\left(\frac{p^2}{q^2}\right)}{(\frac{2p^2}{q^2}-1)}=\frac{p^2}{2p^2-q^2}$$

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