Question

assignment 9.1: trigonometric identities
score: 17/20 answered: 13/16
question 14
simplify \\( \frac{\cot (t)}{\csc (t)-\sin (t)} \\) to a single trig function.
question help: video 1 video 2 message instructor

Explanation:

Step1: Substitute trigonometric identities

Recall that \(\cot(t)=\frac{\cos(t)}{\sin(t)}\), \(\csc(t)=\frac{1}{\sin(t)}\).
So the expression becomes \(\frac{\frac{\cos(t)}{\sin(t)}}{\frac{1}{\sin(t)}-\sin(t)}\).

Step2: Simplify the denominator

Simplify \(\frac{1}{\sin(t)}-\sin(t)=\frac{1 - \sin^{2}(t)}{\sin(t)}\).
By the Pythagorean identity \(\sin^{2}(t)+\cos^{2}(t) = 1\), then \(1-\sin^{2}(t)=\cos^{2}(t)\). So the denominator is \(\frac{\cos^{2}(t)}{\sin(t)}\).

Step3: Simplify the whole expression

The original expression is \(\frac{\frac{\cos(t)}{\sin(t)}}{\frac{\cos^{2}(t)}{\sin(t)}}=\frac{\cos(t)}{\sin(t)}\times\frac{\sin(t)}{\cos^{2}(t)}=\frac{1}{\cos(t)}=\sec(t)\).

Answer:

\(\sec(t)\)