List-I		List-II
a)	Least value of $3{\cos }^2\theta +4{\sin }^2\theta$	p)	$1$
b)	If $A>0,B>0$ and $A+B=\frac{\pi }{3}$ then maximum value of $\tan A\tan B$	q)	$0$
c)	The value of $6\left({\sin }^6\theta +{\cos }^6\theta \right)-9\left({\sin }^4\theta +{\cos }^4\theta \right)+4$	r)	$2$
d)	If $A+B+C=\mathrm{\pi }$ and $\cos A=\cos B\cos C$then  $\tan B\tan C$ value is	s)	$\frac{1}{3}$
		t)	$3$

 

(a) $3{\cos }^2\theta +4{\sin }^2\theta =3{\cos }^2\theta +3{\sin }^2\theta +{\sin }^2\theta$ $=3+{\sin }^2\theta$

Least value of ${\sin }^2\theta =0$

$\therefore$ Least value=3+0=3

(b) A>0, B>0,  $A+B=\frac{\pi }{3}$

Maximum value of $\tan A\tan B$  exits  at $A=B=\frac{\pi }{6}$

$\therefore$Maximum value=$\tan \frac{\pi }{6}\tan \frac{\pi }{6}$=$\frac{1}{3}$

C) $6\left({\sin }^6\theta +{\cos }^6\theta \right)-9\left({\sin }^4\theta +{\cos }^4\theta \right)+4$  Put $\theta =0$

$=6\left(0+1\right)-9\left(0+1\right)+4=6-9+4=1$

(d)  $A+B+C=\pi$

$\cos A=\cos B\cos C$

$\cos \left(\pi -\overline{B}+C\right)=\cos B\cos C$

$-\cos \left(B+C\right)=\cos B\cos C$

$-\cos B\cos C+\sin B\sin C=\cos B\cos C$

$\sin B\sin C=2\cos B\cos C$

$\tan B\tan C=2$