Question

Write the following expression in terms sinθ or cosθ, and then simplify if possible. cscθ − cotθcos(-θ)

Answer 1

Given the expression:

`\csc \theta-\cot \theta\cos (-\theta)`

Let's write the expression in terms of sinθ o cosθ.

To simplify, apply trigonometric identities.

`\begin{gathered} \csc (\theta)=(1)/(\sin\theta) \\ \\ \text{cot}\theta=(\cos \theta)/(\sin \theta) \end{gathered}`

Thus, we have:

`\begin{gathered} \csc \theta-\cot \theta\cos (-\theta) \\ \\ =(1)/(\sin\theta)-(\cos\theta)/(\sin\theta)\cdot\cos (-\theta) \end{gathered}`

Solving further:

Since cos(-θ) is an even function, rewrite it as cos(θ)

`\begin{gathered} (1)/(\sin\theta)-(\cos\theta)/(\sin\theta)\cdot\cos \theta \\ \\ (1)/(\sin\theta)-(\cos \theta\cdot\cos \theta)/(\sin \theta) \\ \\ (1)/(\sin\theta)-(\cos ^2\theta)/(\sin \theta) \\ \\ =(1-\cos ^2\theta)/(\sin \theta) \end{gathered}`

Apply Pythagorean Identity:

`\begin{gathered} (\sin^2\theta)/(\sin\theta) \\ \\ =(\sin \theta\cdot\sin \theta)/(\sin \theta) \\ \\ =\sin \theta \end{gathered}`

ANSWER:

`\sin \theta`