Question

Write the trigonometric expression in terms of sine and cosine, and then simplify.
(sin(theta)-csc(theta)) cos(theta)

Answer 1

look at the brackets 1st

\[\csc(\theta) = \frac{1}{\sin(\theta)}\]

\[\sin(\theta) - \csc(\theta) = \frac{\sin^2(\theta) - 1}{\sin(\theta)}\]

so \[(\sin(\theta) - \csc(\theta))\cos(\theta) = (\frac{\sin^2(\theta) -1}{\sin(\theta)})\cos(\theta)\]

end of of the 1st part

Answer 2

simplifying

\[\sin^2(\theta) - 1 = -(1 - \sin^2(\theta)) = -\cos^2(\theta)\]


so the finally

\[-\cos^2(\theta) \times \frac{\cos( \theta)}{\sin(\theta)} = -\frac{\cos^3(\theta)}{\sin(\theta)}\]