Question

Verify each identity. $1-\frac{\cos ^{2} x}{1+\sin x}=\sin x$

   Verify each identity.
$1-\frac{\cos ^{2} x}{1+\sin x}=\sin x$

Precalculus

Robert Blitzer 5th Edition

Chapter 5, Problem 30

Instant Answer

Solved by Expert Joanna Quigley

03/14/2020

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Step 1: We start with the left hand side of the equation: \[1-\frac{\cos ^{2} x}{1+\sin x}\] Show more…

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Verify each identity. $1-\frac{\cos ^{2} x}{1+\sin x}=\sin x$

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Key Concepts

Algebraic Manipulation

This involves techniques such as factoring, canceling common factors, and rewriting expressions to simplify complex fractions. In the context of the identity verification, algebraic manipulation is applied to transform the left-hand side of the equation step by step until it matches the right-hand side.

Verification of Identities

This is the process of proving that two expressions are equivalent for all permissible values of the variable. Verifying identities requires a systematic approach using known trigonometric and algebraic principles to demonstrate the equality holds universally.

Trigonometric Identities

These are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. In the problem, the Pythagorean identity (sin² x + cos² x = 1) is used to rewrite sin² x as 1 - cos² x in order to simplify and ultimately verify the given identity.

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