91Ó°ÊÓ

We will start by rewriting all the trigonometric functions on both sides of the identity in terms of sine and cosine. Recall that: 1. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) 2. \(\sec \theta = \frac{1}{\cos \theta}\) So, our identity becomes: \(\sin^2 \theta + \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \left(\frac{1}{\cos \theta}\right)^2 - \cos^2 \theta\)

On the left side, we can simplify the tan squared term by squaring the fraction: \(\sin^2 \theta + \frac{\sin^2 \theta}{\cos^2 \theta}\) On the right side, we can simplify the sec squared term by squaring the fraction: \(\frac{1}{\cos^2 \theta} - \cos^2 \theta\)

Now, we want to find a common denominator for both sides of the equation to further simplify them. The common denominator for both sides is \(\cos^2 \theta\). So, our equation will look like: \(\frac{\sin^2 \theta \cdot \cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1 - \cos^4 \theta}{\cos^2 \theta}\)

Recall the trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\) We can rewrite the terms in the numerator of the left side to use this identity: \(\frac{(\sin^2 \theta + \cos^2 \theta) \cdot \sin^2 \theta}{\cos^2 \theta} = \frac{1 - \cos^4 \theta}{\cos^2 \theta}\) Simplify the left side: \(\frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1 - \cos^4 \theta}{\cos^2 \theta}\) Since the denominators are equal, we can now equate the numerators and use our trigonometric identity: \(\sin^2 \theta = 1 - \cos^4 \theta\) On the right side, we can rewrite \(1 - \cos^4 \theta\) as \(1 - (\cos^2 \theta)^2\) and use the identity again to replace \(\cos^2 \theta\) with \(1 - \sin^2 \theta\): \(\sin^2 \theta = 1 - (1 - \sin^2 \theta)^2\) The left and right sides of the equation are equal, so we have successfully proven our identity: \(\sin^2 \theta + \tan^2 \theta = \sec^2 \theta - \cos^2 \theta\)