How do you prove this trig identity? \frac{{sec}^(2) \theta - 1}{ {sec}^(2) \theta } = {sin}^(2) \theta

Question

How do you prove this trig identity?


`\frac{{sec}^(2) \theta - 1}{ {sec}^(2) \theta } = {sin}^(2) \theta`

Answer 1

On the left side, multiply numerator and denominator by
`\cos^2\theta`:

`(\sec^2\theta-1)/(\sec^2\theta)=(\cos^2\theta(\sec^2\theta-1))/(\cos^2\theta\sec^2\theta)=\frac{1-\cos^2\theta}1`

which follows from the fact that
`\sec\theta=\frac1{\cos\theta}`. Then apply the Pythagorean identity,

`\sin^2\theta+\cos^2\theta=1\implies(\sec^2\theta-1)/(\sec^2\theta)=\sin^2\theta`.

and we're done.