Question

How do you verify the identity `sec^4theta=1+2tan^2theta+tan^4theta`?

Answer 1

Proof below

Explanation:

First we will prove `1+tan^2theta = sec^2theta`:
`sin^2theta + cos^2theta = 1`
`sin^2theta/cos^2theta + cos^2theta/cos^2theta = 1/cos^2theta`
`tan^2theta + 1 = (1/costheta)^2`
`1+tan^2theta = sec^2theta`

Now we can prove your question:
`sec^4theta`
`=(sec^2theta)^2`
`=(1+tan^2theta)^2`
`=1+2tan^theta+tan^4theta`