(b) Prove the following identity: \( 1+\tan ^{2} \theta=\frac{1}{\cos ^{2} \theta} \) (2) Hence delermine the values of \( \theta \) to which the identity is undefined State the general solution.

To prove the identity \( 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} \), we can start by recalling the definition of the tangent function in terms of sine and cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Now, squaring both sides gives us: \[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \] Substituting this into the left-hand side of the identity we want to prove: \[ 1 + \tan^2 \theta = 1 + \frac{\sin^2 \theta}{\cos^2 \theta} \] To combine these terms, we can express \( 1 \) as \( \frac{\cos^2 \theta}{\cos^2 \theta} \): \[ 1 + \tan^2 \theta = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \] Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can simplify the numerator: \[ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} \] Thus, we have proven the identity: \[ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} \] ### (2) Determine the values of \( \theta \) for which the identity is undefined The identity \( \frac{1}{\cos^2 \theta} \) is undefined when \( \cos^2 \theta = 0 \). This occurs when \( \cos \theta = 0 \). The cosine function is zero at the following angles: \[ \theta = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \] ### General Solution Thus, the general solution for the values of \( \theta \) for which the identity is undefined is: \[ \theta = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]