If $$\tan (11 \theta) = \cot (7 \theta)$$, then what is the value of $$\sin^2 (6 \theta) + \sec^2 (9 \theta) + \cosec^2 (12 \theta)$$?

Given : $$\tan (11 \theta) = \cot (7 \theta)$$

=> $$\tan (11 \theta) = \tan (90^\circ-7 \theta)$$

=> $$11\theta=90^\circ-7\theta$$

=> $$11\theta+7\theta=18\theta=90^\circ$$

=> $$\theta=5^\circ$$

To find : $$\sin^2 (6 \theta) + \sec^2 (9 \theta) + \cosec^2 (12 \theta)$$

= $$sin^2(30^\circ)+sec^2(45^\circ)+cosec^2(60^\circ)$$

= $$\frac{1}{4}+2+\frac{4}{3}$$

= $$\frac{3+24+16}{12}=\frac{43}{12}$$

=> Ans - (D)