## Proof of Trigonometric Identity### Problem StatementComplete the proof of the identity by choosing the rule that justifies each step.\[(1 - \sin x)(1 + \sin x) = \frac{1}{1 + \tan^{2} x}\]To see a detailed description of a rule, select the "More Information" button to the right of the Rule.### Steps in Proof**Statement | Rule**1. \((1 - \sin x)(1 + \sin x)\)2. \(= 1 - \sin^{2} x\) | Rule ?3. \(= \cos^{2} x\) | Rule ?4. \(= \frac{1}{\sec^{2} x}\) | Rule ?5. \(= \frac{1}{1 + \tan^{2} x}\) | Rule ?### Explanation of Steps and Rules1. The left-hand side expression \((1 - \sin x)(1 + \sin x)\) can be simplified using the difference of squares formula, \(a^2 - b^2\)2. Using the difference of squares formula: \[ a^2 - b^2 = (a - b)(a + b) \] where \(a = 1\) and \(b = \sin x\), we get: \[ 1 - \sin^2 x \]3. According to the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] rearranging this, we get: \[ \cos^2 x = 1 - \sin^2 x \] Therefore: \[ 1 - \sin^2 x = \cos^2 x \]4. Recognizing that \(\sec x = \frac{1}{\cos x}\), we can write: \[ \cos^2 x = \frac{1}{\sec^2 x} \]5. Using another Pythagorean identity for tangent and secant: \[ 1 + \tan^2 x = \sec^2 x \] we can substitute \(\sec^2 x\) with \(1 + \tan^2 x\)

Name: ## Proof of Trigonometric Identity### Problem StatementComplete the p
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: ## Proof of Trigonometric Identity### Problem StatementComplete the proof of the identity by choosing the rule that justifies each step.\[(1 - \sin x)(1 + \sin x) = \frac{1}{1 + \tan^{2} x}\]To see a detailed description of a rule, select the "More