Guided Example #2 - The Pythagorean Identity

What does ${sin}^{2} x - {cos}^{2} x$ equal?

A. $1 + 2 {cos}^{2} x$ B. $2 {sin}^{2} x - 1$ C. $1 - 2 {sin}^{2} x$ D. $1 - {sin}^{2} x$

Solving for ${cos}^{2} x$ in the fundamental trigonometric identity ${sin}^{2} x + {cos}^{2} x = 1$ , gives us ${cos}^{2} x = 1 - {sin}^{2} x$ .

Thus, ${sin}^{2} x - {cos}^{2} x = {sin}^{2} x - (1 - {sin}^{2} x) = 2 {sin}^{2} x - 1$ .

Solving for ${cos}^{2} x$ in the fundamental trigonometric identity ${sin}^{2} x + {cos}^{2} x = 1$ , gives us ${cos}^{2} x = 1 - {sin}^{2} x$ .

Thus, ${sin}^{2} x - {cos}^{2} x = {sin}^{2} x - (1 - {sin}^{2} x) = 2 {sin}^{2} x - 1$ .