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)

$tan\,\theta$

Answer 2

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