Question : If $\left(\frac{\cos A}{1-\sin A}\right)+\left(\frac{\cos A}{1+\sin A}\right)=4$, then what will be the value of $A$? $\left(0^{\circ}<\theta<90^{\circ}\right)$
Option 1: $90^{\circ}$
Option 2: $45^{\circ}$
Option 3: $60^{\circ}$
Option 4: $30^{\circ}$
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Correct Answer: $60^{\circ}$
Solution :
Given: $\left(\frac{\cos A}{1-\sin A}\right)+\left(\frac{\cos A}{1+\sin A}\right) = 4$
⇒ $\frac{\cos A(1+\sin A) + \cos A(1-\sin A)}{(1-\sin A)(1+\sin A)} = 4$
⇒ $\cos A + \cos A \sin A + \cos A - \cos A \sin A = 4(1-\sin^2A)$
⇒ $2\cos A = 4 \cos^2 A$
⇒ $\cos A = \frac{1}{2} =\cos 60^\circ$
⇒ $A = 60^\circ$
Hence, the correct answer is $60^\circ$.
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