Question

Prove that: 2sec^2 theta- sec^4 theta-2cosec^2 theta+cosec^4 theta=cot^4 theta-tan^4 theta

Answer 1

Here $LHS=2\sec^2 \theta-\sec^4 \theta-2\csc^2 \theta+\csc^4 \theta$

Here make use of the identities

$\sec^2 \theta=1+\tan^2 \theta\\ \csc^2 \theta=1+\cot^2 \theta$

Using the above inequalities,

$2\sec^2 \theta-\sec^4 \theta=2(1+\tan^2 \theta)-(1+\tan^2 \theta)^2\\ 2\sec^2 \theta-\sec^4 \theta=2(1+\tan^2 \theta)-(1+2\tan^2+\tan^4 \theta)\\ 2\sec^2 \theta-\sec^4 \theta=1-\tan^4 \theta$

Similarly,

$2\csc^2 \theta-\csc^4 \theta=2(1+\cot^2 \theta)-(1+\cot^2 \theta)^2\\ 2\csc^2 \theta-\csc^4 \theta=2(1+\cot^2 \theta)-(1+2\cot^2+\cot^4 \theta)\\ 2\csc^2 \theta-\csc^4 \theta=1-\cot^4 \theta$

Thus,

$LHS=2\sec^2 \theta-\sec^4 \theta-2\csc^2 \theta+\csc^4\\ LHS=1-\tan^4 \theta -(1-\cot^4 \theta )\\ LHS=\cot^4 \theta-\tan^4 \theta=RHS$

The proof is complete.

Answer 2

Hope this will help you

Mark me brainliest

Follow me..