Question

prove that sin power 4 theta + cos power 4 theta by 1 - 2 sin square theta cos square theta is equals to 1​

Answer 1

I hope my answer is right.

Answer 2

$\frac{sin^4 \theta + cos^4 \theta}{1-2sin^2 \theta cos^2 \theta} = 1$

Solution:

Given that, we have to prove:

$\frac{sin^4 \theta + cos^4 \theta}{1-2sin^2 \theta cos^2 \theta} = 1$

Take the LHS

$\frac{sin^4 \theta + cos^4 \theta}{1-2sin^2 \theta cos^2 \theta}$

Which is,

$\frac{(sin^2 \theta)^2 + (cos^2 \theta)^2}{1-2sin^2 \theta cos^2 \theta}$

The above can be rewritten as:

By the algebraic identity,

$a^2+b^2 = (a+b)^2-2ab$

Therefore,

$\frac{(sin^2 \theta + cos^2 \theta)^2 - 2sin^2 \theta cos^2 \theta}{1-2sin^2 \theta cos^2 \theta}$

We know that,

$sin^2 \theta +cos^2 \theta = 1$

Thus, we get,

$\frac{1 - 2sin^2 \theta cos^2 \theta}{1-2sin^2 \theta cos^2 \theta}\\\\= 1$

Therefore,

LHS = RHS

Thus proved

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