Question

show that tan ^2 tita + tan^4 tita = sec^4 tita - sec^2 tita

Answer 1

HEY mate here is your answer.

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Answer 2

Hi friend
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Your answer
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To prove : - tan²θ + tan⁴θ = sec⁴θ - sec²θ

Now,

LHS : -

tan²θ + tan⁴θ

= sin²θ/cos²θ + sin⁴θ/cos⁴θ

= (sin²cos²θ + sin⁴θ)/cos⁴θ

=> [sin²θ(cos²θ + sin²θ)]/cos⁴θ

=> (sin²θ × 1)/cos⁴θ. [Because , sin²θ + cos²θ = 1 ]

=> sin²θ/cos⁴θ

Again ,

RHS : -

sec⁴θ - sec²θ

= 1/cos⁴θ - 1/cos²θ

= (1 - cos²θ)/cos⁴θ

= sin²θ/cos⁴θ. [Because , 1 - cos²θ = sin²θ]

Therefore,

tan²θ + tan⁴θ = sin²θ/cos⁴θ

Also,

sec⁴θ - sec²θ = sin²θ/cos⁴θ

So,

tan²θ + tan⁴θ = sec⁴θ - sec²θ

Hence, proved.

HOPE IT HELPS