Question

\[ x\sin45^\circ\cos^260^\circ = \frac{\tan^260^\circ\cosec30^\circ} {\sec45^\circ\cot^230^\circ} \]

then \(x=\)

(a) \(2\)
(b) \(4\)
(c) \(8\)
(d) \(16\)

Solution

Using standard values,

\[ \sin45^\circ=\frac1{\sqrt2}, \quad \cos60^\circ=\frac12 \]

\[ \tan60^\circ=\sqrt3, \quad \cosec30^\circ=2 \]

\[ \sec45^\circ=\sqrt2, \quad \cot30^\circ=\sqrt3 \]

Substituting,

\[ x\left(\frac1{\sqrt2}\right)\left(\frac12\right)^2 = \frac{(\sqrt3)^2\times2} {\sqrt2\times(\sqrt3)^2} \]

\[ x\left(\frac1{4\sqrt2}\right) = \frac{3\times2}{\sqrt2\times3} \]

\[ x\left(\frac1{4\sqrt2}\right) = \frac2{\sqrt2} \]

\[ x = \frac2{\sqrt2}\times4\sqrt2 \]

\[ x=8 \]

Answer

\[ \boxed{8} \]

Correct Option: (c)