Problem 31 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

After the step by step breakdown, it's been verified that \(\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}=2 \sec x\) is indeed a valid trigonometric identity.

Step 1: Simplify each fraction

Firstly, keep in mind that the main goal is to simplify the equation into the form 2 sec x. Start by breaking the equation into two fractions: cos x/(1-sin x) and (1-sin x)/cos x.

Step 2: Express the fractions as a common fraction

To add fractions, they should have the same denominator. Thus, multiply the first fraction by cos x/cos x and the second fraction by (1-sin x)/(1-sin x), Thus, the equation will be: (cos^2x)/((1-sin x)cos x) + ((1-sin x)^2)/((1-sin x)cos x).

Step 3: Sum the two fractions

Add the two fractions to get a single fraction. This will yield: [cos^2x + (1 - 2sin x + sin^2x)]/((1-sin x)cos x) = (1+ sin^2x - 2sinx)/((1-sin x)cos x).

Step 4:Substitute sin^2x = 1 - cos^2x

Using the Pythagorean identity (sin^2 x + cos^2 x = 1 or sin^2 x = 1 - cos^2 x), substitute sin^2 x in the equation to get: (1 + 1 - cos^2x - 2sinx)/((1-sin x)cos x) = (2 - cos^2x - 2sinx)/((1-sin x)cos x).

Step 5: Factor

Factorize the equation to get: 2(1 - cosx + sinx)/((1-sin x)cos x) = 2{(1-cosx)(1+sin x)}/((1-sin x)cos x). The (1-sin x) cancel out to leave: 2/(cosx).

Step 6:Convert to sec x

Finally, remember that sec x = 1/cos x. This leads to: 2 sec x. Which matches the right side of the original equation.