Question

Taking \(\theta = 30^\circ\) to verify the following Trigonometric identities:
(i) \(\sin^2\theta + \cos^2\theta = 1\)
(ii) \(1 + \tan^2\theta = \sec^2\theta\)
(iii) \(1 + \cot^2\theta = \csc^2\theta\)

Hint:

To "verify" an identity for a specific value, you must calculate the LHS and RHS separately and then show they are equal. Do not start by assuming they are equal and manipulating the equation. A solid knowledge of the trigonometric values for standard angles (0, 30, 45, 60, 90) is essential.

Answer 1

Step 1: Understanding the Concept:
We need to verify that the three fundamental Pythagorean trigonometric identities hold true for the specific angle \(\theta = 30^\circ\). This involves substituting the known trigonometric values for 30\(^\circ\) into the equations and showing that the Left Hand Side (LHS) equals the Right Hand Side (RHS).

Step 2: Key Formula or Approach:
We need the standard trigonometric values for \(\theta = 30^\circ\): \[\begin{array}{rl} \bullet & \text{\(\sin 30^\circ = \frac{1}{2}\)} \\ \bullet & \text{\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)} \\ \bullet & \text{\(\tan 30^\circ = \frac{1}{\sqrt{3}}\)} \\ \bullet & \text{\(\csc 30^\circ = 2\)} \\ \bullet & \text{\(\sec 30^\circ = \frac{2}{\sqrt{3}}\)} \\ \bullet & \text{\(\cot 30^\circ = \sqrt{3}\)} \\ \end{array}\]

Step 3: Detailed Explanation:
(i) Verify \(\sin^2\theta + \cos^2\theta = 1\) LHS = \(\sin^2(30^\circ) + \cos^2(30^\circ)\) \[ = \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 \] \[ = \frac{1}{4} + \frac{3}{4} = \frac{1+3}{4} = \frac{4}{4} = 1 \] RHS = 1. Since LHS = RHS, the identity is verified.
(ii) Verify \(1 + \tan^2\theta = \sec^2\theta\) LHS = \(1 + \tan^2(30^\circ)\) \[ = 1 + \left(\frac{1}{\sqrt{3}}\right)^2 = 1 + \frac{1}{3} = \frac{3+1}{3} = \frac{4}{3} \] RHS = \(\sec^2(30^\circ)\) \[ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \] Since LHS = RHS, the identity is verified.
(iii) Verify \(1 + \cot^2\theta = \csc^2\theta\) LHS = \(1 + \cot^2(30^\circ)\) \[ = 1 + (\sqrt{3})^2 = 1 + 3 = 4 \] RHS = \(\csc^2(30^\circ)\) \[ = (2)^2 = 4 \] Since LHS = RHS, the identity is verified.

Step 4: Final Answer:
All three trigonometric identities are successfully verified for \(\theta = 30^\circ\).