Question

1. \csc x - \cos x \cot x

Explanation:

Step1: Recall trigonometric identities

Recall that \(\csc x=\frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substitute these into the expression:
\(\csc x-\cos x\cot x=\frac{1}{\sin x}-\cos x\cdot\frac{\cos x}{\sin x}\)

Step2: Simplify the expression

Combine the two terms over a common denominator \(\sin x\):
\(\frac{1 - \cos^{2}x}{\sin x}\)
Recall the Pythagorean identity \(1-\cos^{2}x=\sin^{2}x\). Substitute this into the numerator:
\(\frac{\sin^{2}x}{\sin x}\)

Step3: Cancel common factors

Cancel out one \(\sin x\) from the numerator and the denominator:
\(\sin x\)

Answer:

\(\sin x\)