Question

The value of $\cot {16}^o\cot {44}^o+\cot {44}^o\cot {76}^o-\cot {76}^o\cot {16}^o$ is

• A. $3$
• B. $\frac{1}{3}$
• C. $-\frac{1}{3}$
• D. $-3$

Answer 1

The correct option is A $3$

The value of $\cot 16\cot 44+\cot 44\cot 76-\cot 76\cot 16$

$=(\cot 16\cot 44-1)+(\cot 44\cot 76-1)-(\cot 76\cot 16+1)+3$

$=\frac{\cos (44+16)}{\sin 16sin44}+\frac{\cos (44+76)}{\sin 44\sin 76}-\frac{\cos (76-16)}{\sin 76\sin 16}+3$

$=\frac{\cos 60}{\sin 16\sin 44}+\frac{\cos 120}{\sin 44\sin 76}-\frac{\cos 60}{\sin 76\sin 16}+3$

$=\frac{\cos 60}{\sin 16}\left[\frac{\sin 76-\sin 44}{\sin 76\sin 44}\right]+\frac{\cos 120}{\sin 44\sin 76}+3$

$=\frac{\cos 60\times 2\sin 16\cos 60}{\sin 16\sin 44\sin 76}+\frac{\cos 120}{\sin 44\sin 76}+3$

$=\frac{2\cos 120+1}{\sin 44\sin 76}+3$

$=3$