Column -I		Column -II
a	The maximum value of $\{\cos (2A+\theta )+\cos (2B+\theta \left)\right\}$ is (where A, B are constants)	p	2sin (A + B)
b	The maximum value of $\{\cos 2A+\cos 2B\}$ is (where (A + B) is constant and $A,B\in (0,\pi /2)$)	q	2sec (A + B)
c	The minimum value of $\{\sec 2A+\sec 2B\}$ is  (where (A + B) is constant and $A,B\in (0,\pi /4)$)	r	2cos (A + B)
d	The minimum value of $\sqrt{\{\tan \theta +\cot \theta -2\cos 2(A+B\left)\right\}}$ is (where A, B are constants and $\theta \in (0,\pi /2)$)	s	2cos (A – B)

a)
$\begin{array}{r}\{\cos (2A+\theta )+\cos (2B+\theta \left)\right\}=2\cos (A-B)\cos (A+B+\theta )\end{array}$
Maximum value is 2cos(A–B) when $\cos (A+B+\theta )=1$
b)
$\{\cos 2A+\cos 2B\}=2\cos (A+B)\cos (A-B)$
Maximum value is 2cos(A–B) when cos(A+B) =1

c)For $y=\sec x,x\in (0,\pi /2)$ , tangent drawn to it at
any point lies completely below the graph of y=secx;
thus, $\frac{\sec 2A+\sec 2B}{2}\ge \sec (A+B)$
or $\sec 2A+\sec 2B\ge \sec (A+B)$
Hence, the minimum value is 2sec(A+B).
d)
$\begin{array}{l}\sqrt{\{\tan \theta +\cot \theta -2\cos 2(A+B\left)\right\}}\\ =\sqrt{(\sqrt{\tan \theta }-\sqrt{\cot \theta }{)}^2+2-2\cos 2(A+B)}\\ =\sqrt{(\sqrt{\tan \theta }-\sqrt{\cot \theta }{)}^2+4{\sin }^2(A+B)}\end{array}$
Minimum value occurs when $\sqrt{\tan \theta }=\sqrt{\cot \theta }$ and
minimum value is $\sqrt{4{\sin }^2(A+B)}=2\sin (A+B)$