Question

Prove that tan²theta + cot²theta +2 = sec²theta+cosec²theta
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Answer 1

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To prove :

2 sec²Φ - sec⁴Φ - 2 cosec² Φ + cosec⁴Φ =cot⁴Φ - tan⁴Φ

Proof :

here remember that,

1 - tan²Φ = sec²Φ And

1 - cot²Φ = cosec² Φ

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LHS = 2 sec²Φ - sec⁴Φ- 2 cosec ² Φ + cosec⁴ Φ

=)) sec²Φ ( 2 - sec²Φ ) - cosec²Φ (2 - cosec² Φ )

=)) sec²Φ ( 1 +( 1 - sec²Φ) ) - cosec²Φ ( 1 + (1 - cosec²Φ))

=)) sec²Φ ( 1 + ( -tan²Φ )) - cosec²Φ (1 + (- cot²Φ ))

=)) ( 1+ tan²Φ ) (1 - tan²Φ ) - (( 1 + cot²Φ )(1 - cot² Φ ))

=)) ((1)⁴ - (tan⁴)) -((1)⁴ - (cot⁴))

=)) ( 1 - tan⁴ - 1 + cot⁴ )

=)) -tan⁴ + cot⁴

=)) cot⁴ - tan⁴

=)) RHS

Hence proved

Hope this helps☺✌