Question

Use trigonometric identities to simplify each expression. (1 + cos(x))(1 – cos(x))

Answer 1

sin²(x)

Explanation:

• We can use a trigonometric identity to simplify the expression
  `\sf (1 + \cos(x))(1 - \cos(x))`
• Let's start by applying the identity
  `(a^2 - b^2) = (a + b)(a - b).`

In this case,
`(a = 1\:)\: and\: (b = cos(x)):`

`(1 + \cos(x))(1 - \cos(x)) = 1^2 - \cos^2(x)`

• Now, recall the Pythagorean identity:

`\cos^2(x) + \sin^2(x) = 1`

• Rearranging this gives us

`\cos^2(x) = 1 - \sin^2(x)`

• Substitute
  `\cos^2(x) = 1 - \sin^2(x)` into the expression:

`1^2 - \cos^2(x) \\\\ 1 - (1 - \sin^2(x)) \\\\ 1 - 1 + \sin^2(x) \\\\ \sin^2(x)`

So,
`(1 + \cos(x))(1 - \cos(x))\:\sf simplifies\: to\: \sin^2(x)`