Questions: What does (1/(cos x+1))-(1/(cos x-1)) equal? A. 2/(-sin^2 x) B. 2 csc^2 x C. 1 D. -sin^2 x

Solution Steps

To solve the given expression, we need to find a common denominator and simplify the resulting expression. We will use trigonometric identities to simplify the expression further.

1. Find a common denominator for the two fractions.
2. Combine the fractions into a single fraction.
3. Simplify the resulting expression using trigonometric identities.

We start with the expression:

\[ \frac{1}{\cos x + 1} - \frac{1}{\cos x - 1} \]

To combine these fractions, we find a common denominator, which is \((\cos x + 1)(\cos x - 1)\). Thus, we rewrite the expression as:

\[ \frac{(\cos x - 1) - (\cos x + 1)}{(\cos x + 1)(\cos x - 1)} \]

Now, we simplify the numerator:

\[ (\cos x - 1) - (\cos x + 1) = \cos x - 1 - \cos x - 1 = -2 \]

So, the expression becomes:

\[ \frac{-2}{(\cos x + 1)(\cos x - 1)} \]

Next, we recognize that \((\cos x + 1)(\cos x - 1) = \cos^2 x - 1 = -\sin^2 x\). Therefore, we can rewrite the expression as:

\[ \frac{-2}{-\sin^2 x} = \frac{2}{\sin^2 x} \]

This can also be expressed using the cosecant function:

\[ 2 \csc^2 x \]

Final Answer