Question

1+cos^22x=2(cos^4x+sin^4x)​

Answer 1

$1+cos^2\ 2x = 2(cos^4x + sin^4x)$

Solution:

Given that, we have to prove

$1+cos^2\ 2x = 2(cos^4x + sin^4x)$

Take the RHS

$2(cos^4x + sin^4x)$

Which can be written as

$2((cos^2x)^2 + (sin^2x)^2)$

Use the following identity,

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

Therefore,

$2((sin^2x + cos^2 x)^2 - 2sin^2xcos^2x)\\\\We\ know\ that\ sin^2x + cos^2x = 1\\\\Therefore\\\\2(1 - 2sin^2xcos^2x)\\\\Which\ is\\\\2(1 - \frac{1}{2}(sin2x)^2)\\\\We\ know\ that\ (sin2x)^2 = 1-cos^22x\\\\Therefore\\\\2(1 - \frac{1}{2}(1-cos^2 2x))\\\\Simplify\\\\2 ( 1 - \frac{1}{2} + \frac{1}{2}cos^2 2x)\\\\2 - 1 + cos^2 2x\\\\1 + cos^22x$

Thus LHS = RHS

Thus proved

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Answer 2

Step-by-step explanation:

So ,

LHS = RHS

I hope this will help u