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
$cosec\left(90^{\circ}+\theta\right)+xcos\theta\,cot\left(90^{\circ}+\theta\right)=sin\left(90^{\circ}+\theta\right)$
$\Rightarrow sec\theta+xcos\theta\left(-tan\theta\right)=cos\theta$
$\Rightarrow sec\theta-x\,sin\theta=cos\theta$
$\Rightarrow xsin\theta=sec\theta-cos\theta=\frac{1}{cos\,\theta}-cos\theta$
$\Rightarrow xsin\theta=\frac{1-cos^{2}\,\theta}{cos\,\theta}=\frac{sin^{2}\,\theta}{cos\,\theta}$
$\Rightarrow x=tan\theta$.