Asked in: March 2025 board paperTaking θ = 30°, verify the following trigonometric identities:(i) \(\sin^2\theta + \cos^2\theta = 1\) (ii) \(1 + \tan^2\theta = \sec^2\theta\) (iii) \(1 + \cot^2\theta = \operatorname{cosec}^2\theta\)

Answer: All three identities are verified for θ = 30°.

For θ = 30°: \(\sin 30^\circ = \dfrac{1}{2}\), \(\cos 30^\circ = \dfrac{\sqrt3}{2}\), \(\tan 30^\circ = \dfrac{1}{\sqrt3}\), \(\cot 30^\circ = \sqrt3\), \(\sec 30^\circ = \dfrac{2}{\sqrt3}\), \(\operatorname{cosec} 30^\circ = 2\).

(i) LHS \(= \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{\sqrt3}{2}\right)^2 = \dfrac{1}{4} + \dfrac{3}{4} = 1\) = RHS.

(ii) LHS \(= 1 + \left(\dfrac{1}{\sqrt3}\right)^2 = 1 + \dfrac{1}{3} = \dfrac{4}{3}\); RHS \(= \left(\dfrac{2}{\sqrt3}\right)^2 = \dfrac{4}{3}\). LHS = RHS.

(iii) LHS \(= 1 + (\sqrt3)^2 = 1 + 3 = 4\); RHS \(= 2^2 = 4\). LHS = RHS.