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Question Number 31016 by 78987 last updated on 02/Mar/18

$$showthat1/\mathrm{cosec}x-cotx+1/\mathrm{cosec}x+\cot x=2\mathrm{cosec}x$$

Answered by iv@0uja last updated on 02/Mar/18

$$\frac{1}{\mathrm{cosec}x-\cot x}+\frac{1}{\mathrm{cosec}x+\cot x}$$$$=\frac{1}{\frac{1}{\sin x}-\frac{\cos x}{\sin x}}+\frac{1}{\frac{1}{\sin x}+\frac{\cos x}{\sin x}}$$$$=\frac{\sin x}{1-\cos x}+\frac{\sin x}{1+\cos x}$$$$=\frac{(1+\cos x)\sin x+(1-\cos x)\sin x}{1-{\cos }^{2}x}$$$$=\frac{2\sin x}{{\sin }^{2}x}=\frac{2}{\sin x}=2\mathrm{cosec}x$$

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