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Question Number 176164 by HeferH last updated on 14/Sep/22

$$Proofthat:$$$$$$$$\sqrt{\frac{1-\cos x}{1+\cos x}}+\sqrt{\frac{1+\cos x}{1-\cos x}}=2\cdot \mathrm{cosec}x$$$$$$

Answered by Rasheed.Sindhi last updated on 14/Sep/22

$$\sqrt{\frac{1-\cos x}{1+\cos x}}+\sqrt{\frac{1+\cos x}{1-\cos x}}=2\cdot \mathrm{cosec}x$$$$\mathrm{LHS}:\frac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}+\frac{\sqrt{1+\cos x}}{\sqrt{1-\cos x}}$$$$=\frac{1-\cancel{\cos x}+1+\cancel{\cos x}}{\sqrt{1-{\cos }^{2}x}}$$$$=\frac{2}{\sin x}$$$$=2\cdot \frac{1}{\sin x}$$$$=2\mathrm{cosec}x=\mathrm{RHS}$$

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