Questions: Establish the identity. sec u - tan u = (cos u) / (1 + sin u) (1 - sin u) / (cos u) Rewrite the expression from the previous step by multiplying the numerator and denominator by 1 + sin ( ) / (cos u(1 + sin u)) (Simplify your answer.) Write the numerator of the expression from the previous step in terms of cosine. ( ) / (cos u(1 + sin u))

Solution Steps

To establish the identity, we need to manipulate the given trigonometric expressions to show that they are equivalent. We start by expressing \(\sec u\) and \(\tan u\) in terms of sine and cosine. Then, we simplify the expression \(\sec u - \tan u\) and show that it equals \(\frac{\cos u}{1 + \sin u}\). Next, we multiply the numerator and denominator of the expression by \(1 + \sin u\) to simplify further. Finally, we rewrite the numerator in terms of cosine.

We start with the expression \( \sec u - \tan u \). Rewriting this in terms of sine and cosine gives us: \[ \sec u = \frac{1}{\cos u}, \quad \tan u = \frac{\sin u}{\cos u} \] Thus, we have: \[ \sec u - \tan u = \frac{1}{\cos u} - \frac{\sin u}{\cos u} = \frac{1 - \sin u}{\cos u} \] We need to show that this is equal to \( \frac{\cos u}{1 + \sin u} \).

Next, we simplify the expression \( \frac{1 - \sin u}{\cos u} \) and compare it with \( \frac{\cos u}{1 + \sin u} \). We find that: \[ \frac{1 - \sin u}{\cos u} = \frac{\cos u}{1 + \sin u} \] This confirms that the identity holds.

Now, we multiply the numerator and denominator of \( \frac{1 - \sin u}{\cos u} \) by \( 1 + \sin u \): \[ \frac{(1 - \sin u)(1 + \sin u)}{\cos u(1 + \sin u)} = \frac{1 - \sin^2 u}{\cos u(1 + \sin u)} \] Using the Pythagorean identity \( 1 - \sin^2 u = \cos^2 u \), we can rewrite the numerator: \[ \frac{\cos^2 u}{\cos u(1 + \sin u)} = \frac{\cos u}{1 + \sin u} \]

The numerator in terms of cosine is \( \cos^2 u \). Therefore, the expression simplifies to: \[ \frac{\cos^2 u}{\cos u(1 + \sin u)} \]

Final Answer