Question

rewrite \\(h(u) = \csc u - \cos u \cdot \cot u\\) as an expression in which \\(\sin u\\) appears once and no other trigonometric functions are involved.

hint: watch this video explanation and pause the video along the way as you work.

\\(\bigcirc\\) \\(\sin^2 u\\)
\\(\bigcirc\\) \\(\sin u\\)
\\(\bigcirc\\) \\(1 + \sin u\\)
\\(\bigcirc\\) \\(\frac{1}{\sin u}\\)

Explanation:

Express in terms of sine and cosine

Using the Trigonometric Identities knowledge point

$$ \csc u = \frac{1}{\sin u}, \quad \cot u = \frac{\cos u}{\sin u} $$

Substitute into the function

Using the Trigonometric Identities knowledge point

$$ h(u) = \frac{1}{\sin u} - \cos u \cdot \frac{\cos u}{\sin u} = \frac{1}{\sin u} - \frac{\cos^2 u}{\sin u} $$

Combine terms and simplify

Using the Trigonometric Identities knowledge point

$$ h(u) = \frac{1 - \cos^2 u}{\sin u} = \frac{\sin^2 u}{\sin u} = \sin u $$

Answer:

• (A) \(\sin^2 u\)
• (B) \(\sin u\) (Correct answer)
• (C) \(1 + \sin u\)
• (D) \(\frac{1}{\sin u}\)