Question

Find $x$ from the equation:
$cosec \,\left(90^{\circ}+\theta\right)+x\, \cos \,\theta \,\cot \left(90^{\circ}+\theta\right)=\sin \left(90^{\circ}+\theta\right)$

• A. $\cot \,\theta$
• B. $\tan \,\theta$
• C. $-\tan \,\theta$
• D. $-\cot \,\theta$

Answer 1

Answer: (B)

The given equation is
$cosec\,\,\left(90^{\circ}+\theta\right)+x \cos \theta \cot \left(90^{\circ}+\theta\right)=\sin \left(90^{\circ}+\theta\right)$
$\Rightarrow \sec \theta+x \cos \theta(-\tan \theta)=\cos \theta$
$\Rightarrow \sec \theta-x \sin \theta=\cos \theta$
$\Rightarrow x \sin \theta=\sec \theta-\cos \theta=\frac{1}{\cos \theta}-\cos \theta$
$\Rightarrow x \sin \theta=\frac{1-\cos ^{2} \theta}{\cos \theta}=\frac{\sin ^{2} \theta}{\cos \theta} \Rightarrow x=\tan \theta$