Question

simplify cot(t)/csc(t)-sin(t)

Answer 1

ANSWER

= sec(t)

EXPLANATION

First let's rewrite this in terms of sines and cosines:

$\frac{\cot t}{\csc t-\sin t}$

The cotangent is the reciprocal of the tangent:

$\cot t=\frac{1}{\tan t}=\frac{\cos t}{\sin t}$

And the cosecant is the reciprocal of the sine:

$\csc t=\frac{1}{\sin t}$

Replace into the given expression:

$\frac{\cot t}{\csc t-\sin t}=\frac{\frac{\cos t}{\sin t}}{\frac{1}{\sin t}-\sin t}$

We can add the two terms in the denominator:

$\frac{\frac{\cos t}{\sin t}}{\frac{1}{\sin t}-\sin t}=\frac{\frac{\cos t}{\sin t}}{\frac{1-\sin ^2t}{\sin t}}$

The denominators of each fraction get cancelled out:

$\frac{\frac{\cos t}{\sin t}}{\frac{1-\sin^2t}{\sin t}}=\frac{\cos t}{1-\sin ^2t}$

We still can simplify this a little further. Remember the identity:

$\cos ^2t+\sin ^2t=1$

If we solve it for cos²t:

$\cos ^2t=1-\sin ^2t$

We have the same expression of the denominator. So let's replace the denominator by cos²t:

$\frac{\cos t}{1-\sin^2t}=\frac{\cos t}{\cos ^2t}$

Simplify the square:

$\frac{\cos t}{\cos^2t}=\frac{1}{\cos t}$

And this is the secant of t:

$\frac{1}{\cos t}=\sec t$