What is the value of \(\frac{2[\sin(21^\circ)\cos(69^\circ)+\cos(21^\circ)\sin(69^\circ)]+3\sec(54^\circ)\sin(36^\circ)}{\operatorname{cosec}^2(15^\circ)-\tan^2(75^\circ)}\)?

Solution

Solution: Using \(\sin A\cos B+\cos A\sin B=\sin(A+B)\), the bracket becomes \(\sin90^\circ=1\), so the first part is 2. Also, \(\sec54^\circ\sin36^\circ=1\), because \(\cos54^\circ=\sin36^\circ\). Thus, the numerator is \(2+3=5\). For the denominator, \(\tan75^\circ=\cot15^\circ\), so \(\operatorname{cosec}^2 15^\circ-\tan^2 75^\circ=\operatorname{cosec}^2 15^\circ-\cot^2 15^\circ=1\). The final value is 5. Trick: Use complementary-angle identities and \(\operatorname{cosec}^2 A-\cot^2 A=1\).