Prove the identity of a trigonometric: cos4 x - sin4 x = cos2x

Question

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To prove the identity cos⁴ x − sin⁴ x = cos(2x), we will start with the left-hand side (LHS) and transform it into the right-hand side (RHS). Step 1: Rewrite the LHS using the difference of squares formula, a² − b² = (a−b)(a+b). Here, a = cos² x and b = sin² x. cos⁴ x − sin⁴ x = (cos² x)² − (sin² x)²; = (cos² x − sin² x)(cos² x + sin² x); Step 2: Apply the Pythagorean identity, sin² x + cos² x = 1. (cos² x − sin² x)(1) = cos² x − sin² x; Step 3: Use the double angle identity for cosine, cos(2x) = cos² x − sin² x. cos² x − sin² x = cos(2x); Thus, we have shown that cos⁴ x − sin⁴ x = cos(2x). The identity is proven. The identity is proven.

Prove the identity of a trigonometric: cos4 x - sin4 x = cos2x

Answer

Step-by-step solution

Rewrite the LHS using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ .

Apply the Pythagorean identity, $\sin^2 x + \cos^2 x = 1$ .

Use the double angle identity for cosine, $\cos(2x) = \cos^2 x - \sin^2 x$ .

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Prove the identity of a trigonometric: cos4 x - sin4 x = cos2x

Answer

Step-by-step solution

Rewrite the LHS using the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b).

Apply the Pythagorean identity, sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1sin2x+cos2x=1.

Use the double angle identity for cosine, cos⁡(2x)=cos⁡2x−sin⁡2x\cos(2x) = \cos^2 x - \sin^2 xcos(2x)=cos2x−sin2x.

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Rewrite the LHS using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ .

Apply the Pythagorean identity, $\sin^2 x + \cos^2 x = 1$ .

Use the double angle identity for cosine, $\cos(2x) = \cos^2 x - \sin^2 x$ .