Question

standard 16 homework
score: 11/23 answered: 8/16
question 9
simplify the following expression to a single trig function with no fractions.
\sin x + \cot x \cos x =
question help: vid basic funcs trig

Explanation:

Step1: Recall cotangent identity

Recall that \(\cot x=\frac{\cos x}{\sin x}\). Substitute this into the expression:
\(\sin x + \cot x\cos x=\sin x+\frac{\cos x}{\sin x}\cdot\cos x\)

Step2: Simplify the second term

Simplify \(\frac{\cos x}{\sin x}\cdot\cos x=\frac{\cos^{2}x}{\sin x}\). Now the expression is \(\sin x+\frac{\cos^{2}x}{\sin x}\).

Step3: Combine terms over a common denominator

Combine the two terms over the common denominator \(\sin x\): \(\frac{\sin^{2}x+\cos^{2}x}{\sin x}\).

Step4: Use Pythagorean identity

Recall the Pythagorean identity \(\sin^{2}x + \cos^{2}x = 1\). Substitute this into the numerator: \(\frac{1}{\sin x}\).

Step5: Recall reciprocal identity

Recall that \(\frac{1}{\sin x}=\csc x\).

Answer:

\(\csc x\)