Correct Answer – Option 2 : \(\rm \cot \left( \frac {\pi} 4 – \frac{A}{2} \right)\)

Concept:

• sin 2x = 2 sin x cos x
• cos 2x = cos² x – sin² x

Calculation:

Consider, tan A + sec A.

= \(\rm \frac {\sin A}{\cos A} + \frac {1}{\cos A}\)

= \(\rm \frac {\sin A+1}{\cos A}\)

= \(\rm \frac {2\sin \frac{A}{2}\cos \frac{A}{2}+\sin^2\frac{A}{2}+\cos^2\frac{A}{2}}{\cos^2 \frac{A}{2}-\sin^2\frac{A}{2}}\)

= \(\rm \frac {\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)^2}{\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)\left(\cos \frac{A}{2}-\sin \frac{A}{2}\right)}\)

= \(\rm \frac {\sin \frac{A}{2}+\cos \frac{A}{2}}{\cos \frac{A}{2}-\sin \frac{A}{2}}\)

Dividing by \(\rm \sin\frac{A}{2}\), we get:

= \(\rm \frac {\cot\frac{A}{2}+1}{\cot\frac{A}{2}-1}\)

Using \(\rm \cot\frac{\pi}{4}=1\), it can be written as;

= \(\rm \frac {\cot\frac{\pi}{4}\cot\frac{A}{2}+1}{\cot\frac{A}{2}-\cot\frac{\pi}{4}}\)

= \(\rm \cot \left( \frac {\pi} 4 – \frac{A}{2} \right)\)