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Chapter 5: Problem 29

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

Short Answer

Expert verified
After simplifying and cancelling out identical terms, the left side of the equation equals the right side, \(\cos x\), verifying the identity.

Step by step solution

01

Rewrite the Given Equation

The given equation is \(1 - \frac{\sin^2 x}{1 + \cos x} = \cos x\). In this step, it will be useful to focus on the fraction. Remember that subtracting a fraction is the same as adding a negative fraction, so the identity can be rewritten as \(1 + \frac{-\sin^2 x}{1 + \cos x} = \cos x\).
02

Simplify Negative Fraction

Since numerator and denominator in the fraction both have a negative sign, the fraction itself can be rewritten as: \(1 - \frac{\sin^2 x}{-(1 + \cos x)}\). Now using the Pythagorean Identity we can substitute \(\sin^2 x\) in the equation. \(\sin^2 x = 1 - \cos^2 x\). So the equation becomes \(1 - \frac{1 - \cos^2 x}{-(1 + \cos x)}\)
03

Simplify the Fraction

Multiply the fraction by negative one in the numerator and the denominator of the fraction, converting the fraction into: \(1 - \frac{-1 + \cos^2 x}{1 + \cos x}\). Afterwards, it can be further separated into two separate fractions: \(1 - \frac{-1}{1 + \cos x} + \frac{\cos^2 x}{1 + \cos x}\). This simplifies to \(1 + \frac{1}{1 +\cos x} + \frac{\cos^2 x}{1 + \cos x}\)
04

Obtain the Final Solution

The solution to this equation now become clearer when you remember that \( \frac{1}{1 + \cos x} + \frac{\cos^2 x}{1 + \cos x} = \cos x\). So, the entire equation now simplifies to \(1 + \cos x = \cos x\). Cancelling out similar terms, the resulting equation is identical to the original equation: \(\cos x = \cos x\).

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