Without tables or a calculator, evaluate exactly the difference \(\displaystyle\frac{5\cos 76^\circ}{\sin 14^\circ}-\frac{\sec 26^\circ}{\mathrm{cosec}\,64^\circ}\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4
In each fraction the two angles add up to \(90^\circ\). For complementary angles, \(\cos(90^\circ-A)=\sin A\) and \(\sec(90^\circ-A)=\mathrm{cosec}\,A\), so each fraction has the same value on the top and the bottom.
\(\cos 76^\circ=\cos(90^\circ-14^\circ)=\sin 14^\circ\), so \(\displaystyle\frac{5\cos 76^\circ}{\sin 14^\circ}=5\)
\(\sec 26^\circ=\sec(90^\circ-64^\circ)=\mathrm{cosec}\,64^\circ\), so \(\displaystyle\frac{\sec 26^\circ}{\mathrm{cosec}\,64^\circ}=1\)
Value \(=5-1=4\)