Question

(√cosecA-1\√cosecA+1)+(√cosecA+1\√cosecA-1) =2secA​

Answer 1

Hope it is easy to understand

And 1/cosA = secA. So, 2/cosA = 2secA

Answer 2

To prove : $\sqrt{\frac{\csc A-1}{\csc A +1}}+\sqrt{\frac{\csc A+1}{\csc A -1}}=2sec A$

Explanation:

solving LHS

$\sqrt{\frac{\csc A-1}{\csc A +1}}+\sqrt{\frac{\csc A+1}{\csc A -1}}\\\\\text{rationalsing we get}\\\\\sqrt{\frac{\csc A-1}{\csc A +1}\times \frac{A-1}{A-1}}+\sqrt{\frac{\csc A+1}{\csc A -1}\times \frac{A+1}{A+1}}\\\\\sqrt{\frac{(A-1)^2}{-\cot^2 A}}+\sqrt{\frac{(A+1)^2}{-\cot^2 A}}\\\\\frac{A-1}{\cot A }+\frac{A+1}{\cot A}\\\\\text{seperating }\\\\\frac{\csc A}{\cot A}- \frac{1}{\cot A}+\frac{\csc A}{\cot A}+\frac{1}{\cot A}\\\\\cot A = \frac{\cos A}{\sin A}, \csc A = \frac{1}{\sin A}$

$\\\\\frac{\frac{1}{\sin A}}{\frac{\cos A}{\sin A}}+\frac{\frac{1}{\sin A}}{\frac{\cos A}{\sin A}}\\\\\frac{1}{\cos A}+\frac{1}{\cos A}\\\\\sec A +\sec A \\\\2sec A$

= RHS

hence proved

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