Solution

For this demonstration, we will start with the left hand side and show that we can derive the right hand side using algebra and trig identities: \[ \begin{array}{rlrr} &\sin^2\theta +\sin^2\theta\tan^2\theta\\=&\sin^2\theta(1+\tan^2\theta)&\text{Algebra (Factoring)}\\ =&\sin^2\theta(\sec^2\theta)&\text{Pythagorean Identity (Secant)}\\ =&\sin^2\theta\times\frac{1}{\cos^2\theta}&\text{Definition of Secant}\\ =&\left(\dfrac{\sin\theta}{\cos\theta}\right)^2&\text{Algebra}\\ =&\tan^2\theta&\text{Definition of Tangent} \end{array} \] Notice that in this example I did not replace the Tangent with Sines and Cosines. Instead, I used the Secant version of the Pythagorean identity to more efficiently demonstrate my steps. Technically, this can be demonstrate in either way.