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Problem 23
23
☆
Prove the identity
\(\displaystyle \frac{\cos x}{1+\sin x}+\frac{\cos x}{1-\sin x}\equiv 2\sec x.\)
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Solution
\[ \frac{\cos x(1-\sin x)+\cos x(1+\sin x)}{1-\sin^2x} =\frac{2\cos x}{\cos^2x}=\frac{2}{\cos x}=2\sec x. \]
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