Question

if cot⁴∅-cot²∅=1 then show that sec⁴∅-sec²∅= 1

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Answer 1

Answer: LHS

= 2sec² θ – sec⁴ θ – 2 cosec²θ + cosec⁴ θ

= sec² θ (2 – sec² θ) – cosec²θ (2 - cosec² θ)

Now we will replace sec² θ by 1 + tan² θ and cosec² θ by 1+ cot² θ

= (1+tan² θ) (2 –1-tan² θ) – (1+cot² θ)(2 - 1-cot² θ)

=(1+tan² θ)(1-tan² θ)-(1+cot² θ)(1-cot² θ)

=(1-tan⁴ θ) - (1-cot⁴ θ)

=cot⁴ θ -tan⁴ θ

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