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