tan4t+tan2t=sec4t-sec2t

Question

tan⁴t+tan²t=sec⁴t-sec²t

Answer 1

Here is another way:

tan⁴t+tan²t

= tan²t(tan²t+1)

= (sec²t-1)(sec²t), since tan²t+1 = sec²t

= sec⁴t-sec²t

Answer 2

tan⁴t+tan²t=sec⁴t-sec²t

 

The left side can be written as (tan²t)(tan²t) + tan²t

 

Using the identity tan²θ = sec²θ - 1, I did some substitution in the first step and then the rest is expanding the expression and completing the addition and subtraction.

 

 

(sec²θ-1)(sec²θ-1) + (sec²θ-1)

 

sec⁴θ - sec²θ - sec²θ + 1 + sec²θ - 1

 

sec⁴θ - sec²θ