Question

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

Answer 1

= sec(t)

Step-by-step explanation

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

`(\cot t)/(\csc t-\sin t)`

The cotangent is the reciprocal of the tangent:

`\cot t=(1)/(\tan t)=(\cos t)/(\sin t)`

And the cosecant is the reciprocal of the sine:

`\csc t=(1)/(\sin t)`

Replace into the given expression:

`(\cot t)/(\csc t-\sin t)=((\cos t)/(\sin t))/((1)/(\sin t)-\sin t)`

We can add the two terms in the denominator:

`((\cos t)/(\sin t))/((1)/(\sin t)-\sin t)=((\cos t)/(\sin t))/((1-\sin ^2t)/(\sin t))`

The denominators of each fraction get cancelled out:

`((\cos t)/(\sin t))/((1-\sin^2t)/(\sin t))=(\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:

`(\cos t)/(1-\sin^2t)=(\cos t)/(\cos ^2t)`

Simplify the square:

`(\cos t)/(\cos^2t)=(1)/(\cos t)`

And this is the secant of t:

`(1)/(\cos t)=\sec t`