(iii) \( \sec ^{2} \theta-\cos ^{2} \beta=\sin ^{2} \beta+\tan ^{2} \theta \)

To simplify the equation \( \sec^2 \theta - \cos^2 \beta = \sin^2 \beta + \tan^2 \theta \), we can start by recalling some trigonometric identities. 1. The identity for secant is: \[ \sec^2 \theta = 1 + \tan^2 \theta \] 2. The identity for tangent is: \[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \] 3. The Pythagorean identity states: \[ \sin^2 \beta + \cos^2 \beta = 1 \] Now, let's substitute the identity for \( \sec^2 \theta \) into the equation: \[ 1 + \tan^2 \theta - \cos^2 \beta = \sin^2 \beta + \tan^2 \theta \] Next, we can simplify this equation: 1. Subtract \( \tan^2 \theta \) from both sides: \[ 1 - \cos^2 \beta = \sin^2 \beta \] 2. Now, recall that \( 1 - \cos^2 \beta = \sin^2 \beta \) (from the Pythagorean identity). Thus, we have: \[ \sin^2 \beta = \sin^2 \beta \] This shows that the original equation is indeed an identity, as both sides are equal. Therefore, the equation \( \sec^2 \theta - \cos^2 \beta = \sin^2 \beta + \tan^2 \theta \) holds true for all values of \( \theta \) and \( \beta \) where the functions are defined.