NDA 2026 — Mathematics PYQ

Step 1: Simplify the right-hand side

$$\tan 2\alpha +\cot 2\alpha =\frac{\sin 2\alpha }{\cos 2\alpha }+\frac{\cos 2\alpha }{\sin 2\alpha }=\frac{{\sin }^{2}2\alpha +{\cos }^{2}2\alpha }{\sin 2\alpha \cos 2\alpha }=\frac{1}{\sin 2\alpha \cos 2\alpha }$$

Multiply numerator and denominator by $2$:

$$\frac{2}{2\sin 2\alpha \cos 2\alpha }=\frac{2}{\sin 4\alpha }=2\csc 4\alpha$$

Step 2: Equate to the left-hand side

$$2\sec 4\beta =2\csc 4\alpha ⟹\sec 4\beta =\csc 4\alpha$$

$$\sec 4\beta =\sec \left(\frac{\pi }{2}-4\alpha \right)$$

$$4\beta =\frac{\pi }{2}-4\alpha ⟹4\alpha +4\beta =\frac{\pi }{2}$$

$$4(\alpha +\beta )=\frac{\pi }{2}⟹\alpha +\beta =\frac{\pi }{8}$$

Final Answer:

The possible value of $(\alpha +\beta )$ is $\frac{\pi }{8}$, which corresponds to option (d).