Question

Prove that : 2 sec²theta - sec⁴theta - 2 cosec ² theta + cosec⁴theta =cot⁴theta- tan⁴theta ​

Answer 1

To prove :

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

Proof :

here remember that,

1 - tan²Φ = sec²Φ And

1 - cot²Φ = cosec² Φ

_____________________________

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 .

Answer 2

Step-by-step explanation:

L H S

=2 sec²theta- sec⁴theta- 2 cosec²theta+ cosec ⁴theta

=sec ²theta(2-sec² theta)- cosec²theta (2-cosec² theta)

now we will replace sec²theta by 1 +tan²theta and cosec² theta by 1+cot²theta

=(1+tan²theta)(2-1-tan²theta)-(1+cot²theta)(2-1-cot²theta)

=(1+tan²theta)(1-tan²theta)-(1+cot²theta)(1-cot²theta)

=(1-tan⁴theta)-(1-cot⁴theta)

=cot⁴theta-tan⁴theta