Match the following 

	Column I		Column II
A	If  $\tan \theta$  is the G.M between  $\sin \theta$  And  $\cos \theta$  then $1-4{\sin }^3\theta +3{\sin }^4\theta -{\sin }^4\theta =$	P	1
B	$\sqrt{3}\cot {20}^{\circ }-4\cos {20}^{\circ }=$	Q	0
C	$\cot {16}^{\circ }\cot {44}^{\circ }+\cot {44}^{\circ }\cot {76}^{\circ }-\cot {76}^{\circ }\cot {16}^{\circ }=$	R	3
D	$\sum _{r-1}^9{\sin }^2\left(\frac{r\pi }{18}\right)=$	S	5

$A-P;B-P;C-R;D-S$

A. ${\tan }^2\theta =\sin \theta \cos \theta \Rightarrow \sin \theta ={\cos }^2\theta$
           $\therefore (1-{\sin }^2\theta )+(1-3{\sin }^2\theta )+3{\sin }^4\theta -{\sin }^6\theta$
$={\cos }^2\theta +{(1-{\sin }^2\theta )}^2={\cos }^2\theta +{\cos }^6\theta ={\cos }^2\theta +{\sin }^2\theta =1$
B. $\sin {40}^0=\sin ({60}^0-{20}^0)$
 $2\sin {20}^0\cos {20}^0=\frac{\sqrt{3}}{2}\cos {20}^0-\frac{1}{2}\sin {20}^0$
 $4\cos {20}^0=\sqrt{3}\cot {20}^0-1$
C. $\frac{3+\cot {76}^0\cot {16}^0}{\cot {76}^0+\cot {116}^0}=\frac{3\sin {76}^0\sin {16}^0+\cos {76}^0\cos {16}^0}{\sin ({76}^0+{16}^0)}$
= $\frac{2\sin {76}^0\sin {16}^0+\cos ({76}^0-{16}^0)}{\sin ({76}^0+{16}^0)}$= $\frac{\cos {60}^0-\cos {92}^0+\cos {60}^0}{\sin {92}^0}$= $\frac{1-\cos {92}^0}{\sin {92}^0}$= $\tan {46}^{0 }$
= $cot {44}^0$
D.  ${\sin }^{2n}\left(\frac{\pi }{18}\right)+{\sin }^2\left(\frac{2\pi }{18}\right)+\mathrm{.....}+{\sin }^2\left(\frac{\pi }{2}\right)=5$