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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
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B
1
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C
1
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D
2
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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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