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Chapter 5: Problem 68

Verify the identity. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Short Answer

Expert verified
The given trigonometric identity \(\cos ^{4} x-\sin ^{4} x = \cos 2x\) is valid and has been verified

Step by step solution

01

Rewrite the expression

We start by rewriting \(\cos ^{4} x-\sin ^{4} x\) as \((\cos ^{2} x-\sin ^{2} x)^2\), using the identity for the difference of squares.
02

Apply the trigonometric identity

Next, use the Pythagorean identity to simplify: \(\cos ^{2} x + \sin ^{2} x = 1\). Therefore, \(\cos ^{2} x = 1 - \sin ^{2} x\). Substitute this into our equation to get: \((2\cos ^{2} x-1)^2 = \cos 2x\)
03

Simplify the equation

Now recall that the double-angle identity \(\cos 2x = 2\cos ^{2} x - 1\). Substitute this into the equation from step 2 to get: \( (2\cos ^{2} x - 1) = \cos 2x\)
04

Check if the two sides of the equation are equal

It appears that the both sides of the equations are same and hence \(\cos ^{4} x-\sin ^{4} x = \cos 2x\)

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