How to you simplify cotx+tanx?

2 Answers
Dec 7, 2017

The answer is =2csc(2x)

Explanation:

We need

tanx=sinx/cosx

cotx=cosx/sinx

cos^2x+sin^2s=1

sin2x=2sinxcosx

cscx=1/sinx

Therefore,

cotx+tanx=cosx/sinx+sinx/cosx

=(cos^2x+sin^2x)/(sinxcosx)

=1/(sinxcosx)

=2/sin(2x)

=2csc(2x)

Dec 7, 2017

Alternatively, we can get sec(x)csc(x)

Explanation:

If we write cot(x) as 1/tan(x), we get:
cot(x)+tan(x)=1/tan(x)+tan(x)

Then we bring under a common denominator:
=1/tan(x)+(tan(x)*tan(x))/tan(x)=(1+tan^2(x))/tan(x)

Now we can use the tan^2(x)+1=sec^2(x) identity:
=sec^2(x)/tan(x)

To try and work out some of the relationships between these functions, let's represent the functions in terms of a right triangle. We'll let a be the adjacent, b be the opposite and c be the hypotenuse. This gives:
=((c/a)^2)/(b/a)=c^2/a^2*a/b=c^2/(ab)=c/a*c/b

Now if we convert back to trigonometric functions, we get:
c/a*c/b=sec(x)csc(x)

This is equal to Narad T's answer of 2csc(2x)