Problem 29 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(1 + \frac{-\sin^2 x}{1 + \cos x} = \cos x\).

Step 2: 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 \(1 - \frac{1 - \cos^2 x}{-(1 + \cos x)}\)

Step 3: 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 \(1 + \frac{1}{1 +\cos x} + \frac{\cos^2 x}{1 + \cos x}\)

Step 4: 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: \(\cos x = \cos x\).