Question

# $$\cot ^ { 2 } \theta \left( \dfrac { \sec \theta - 1 } { 1 + \sin \theta } \right) + \sec ^ { 2 } \theta \left( \dfrac { \sin \theta - 1 } { 1 + \sec \theta } \right) =$$

A
0
B
1
C
1
D
2

Solution

## The correct option is A $$0$$Given, $$\cot^{2}\theta \left (\dfrac {\sec \theta -1}{1 + \sin \theta}\right ) + \sec^{2}\theta \left (\dfrac {\sin \theta - 1}{1 + \sec \theta}\right )$$$$=\dfrac {\cos {\theta}^{2}}{\sin {\theta}^{2}} \times \dfrac {\left (\dfrac {1 - \cos \theta}{\cos \theta}\right )}{(1 + \sin \theta)} + \dfrac {1}{\cos^{2}\theta} \left (\dfrac {\sin \theta - 1}{\dfrac {\cos \theta + 1}{\cos \theta}}\right )$$$$=\dfrac {\cos \theta}{(1 - \cos^{2}\theta)} \times \dfrac {(1 - \cos \theta)}{(1 +\sin \theta)} + \dfrac {1}{\cos \theta} \left (\dfrac {\sin \theta - 1}{\cos \theta + 1}\right )$$$$=\dfrac {\cos \theta}{(1 + \cos \theta)} \times \dfrac {1}{(1 + \sin \theta)} + \dfrac {\sin \theta - 1}{\cos \theta (1 + \cos \theta)}$$$$=\dfrac {\cos{\theta}^{2} + (1 + \sin \theta) (\sin \theta - 1)}{(1 + \cos \theta)(1 + \sin \theta) \cos \theta}$$$$=\dfrac {\cos{\theta}^{2} + \sin{\theta}^{2} - 1}{(1 + \cos \theta)(1 + \sin \theta)\cos \theta} = 0$$.Mathematics

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