Question

Prove that :

\[ 3\sin\frac{\pi}{6}\sec\frac{\pi}{3} – 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} = 1 \]

Solution

Using standard values,

\[ \sin\frac{\pi}{6}=\frac12, \qquad \sec\frac{\pi}{3}=2 \]

\[ \sin\frac{5\pi}{6}=\frac12, \qquad \cot\frac{\pi}{4}=1 \]

Substituting these values,

\[ \begin{aligned} &3\sin\frac{\pi}{6}\sec\frac{\pi}{3} – 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} \\[4pt] =& 3\left(\frac12\right)(2) – 4\left(\frac12\right)(1) \\[4pt] =& 3-2 \\[4pt] =& 1 \end{aligned} \]

Hence Proved.