Question

If tan⁴ θ + tan² θ = 1, prove that: cos⁴ + cos² θ = 1

Answer 1

Given:

tan⁴ θ + tan² θ = 1

tan² θ(tan² θ + 1) = 1

Use the Trigonometric Identity:

1 + tan² θ = sec² θ

tan² θ . sec² θ = 1

Convert to sine and cosine:

tan² θ with `sin^2 θ/cos^2 θ` and sec² θ with `1/cos^2 θ`

`sin^2 θ/cos^2 θ . 1/cos^2 θ = 1`

`sin^2 θ/cos^4 θ = 1`

sin² θ = cos⁴ θ

Use the Identity sin² θ = 1 − cos² θ

1 − cos² θ = cos⁴ θ

1 = cos⁴ θ + cos² θ

cos⁴ θ + cos² θ = 1

Hence Proved.