Question

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

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

Answer 1

Answer: (B)

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