Question : $\frac{1+\cos \theta-\sin ^2 \theta}{\sin \theta(1+\cos \theta)} \times \frac{\sqrt{\sec ^2 \theta+\operatorname{cosec}^2 \theta}}{\tan \theta+\cot \theta}, 0^{\circ}<\theta<90^{\circ}$, is equal to:
Option 1: $\sin \theta$
Option 2: $\cos \theta$
Option 3: $\operatorname{cosec} \theta$
Option 4: $\cot \theta$
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Correct Answer: $\cot \theta$
Solution : $\frac{1+\cos \theta-\sin ^2 \theta}{\sin \theta(1+\cos \theta)} \times \frac{\sqrt{\sec ^2 \theta+\operatorname{cosec}^2 \theta}}{\tan \theta+\cot \theta}$ $=\frac{1+\cos \theta-(1-\cos ^2 \theta)}{\sin \theta(1+\cos \theta)} \times \frac{\sqrt{1+\tan ^2 \theta+1+\operatorname{cot}^2 \theta}}{ \tan \theta+\cot \theta}$ $=\frac{\cos \theta+\cos ^2 \theta}{\sin \theta(1+\cos \theta)} \times \frac{\sqrt{(\tan \theta+\operatorname{cot} \theta)^2}}{\tan \theta+\cot \theta}$ $=\frac{\cos \theta(1+\cos \theta)}{\sin \theta(1+\cos \theta)} \times \frac{\tan \theta+\operatorname{cot} \theta}{\tan \theta+\cot \theta}$ $=\cot\theta$ Hence, the correct answer is $\cot \theta$.
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Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:
Question : Let $0^{\circ}<\theta<90^{\circ}$, $\left(1+\cot ^2 \theta\right)\left(1+\tan ^2 \theta\right) × (\sin \theta-\operatorname{cosec} \theta)(\cos \theta-\sec \theta)$ is equal to:
Question : The expression $\frac{(1-\sin \theta+\cos \theta)^2(1-\cos \theta) \sec ^3 \theta\; {\operatorname{cosec}}^2 \theta}{(\sec \theta-\tan \theta)(\tan \theta+\cot \theta)}, 0^{\circ}<\theta<90^{\circ}$, is equal to:
Question : $\frac{(1+\sec \theta \operatorname{cosec} \theta)^2(\sec \theta-\tan \theta)^2(1+\sin \theta)}{(\sin \theta+\sec \theta)^2+(\cos \theta+\operatorname{cosec} \theta)^2}, 0^{\circ}<\theta<90^{\circ}$, is equal to:
Question : The expression $\frac{\cos ^4 \theta-\sin ^4 \theta+2 \sin ^2 \theta+3}{(\operatorname{cosec} \theta+\cot \theta+1)(\operatorname{cosec} \theta-\cot \theta+1)-2}, 0^{\circ}<\theta<90^{\circ}$, is equal to:
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