How to you simplify cotx+tanx?

2 Answers
Dec 7, 2017

The answer is =2csc(2x)

Explanation:

We need

tanx=sinxcosx

cotx=cosxsinx

cos2x+sin2s=1

sin2x=2sinxcosx

cscx=1sinx

Therefore,

cotx+tanx=cosxsinx+sinxcosx

=cos2x+sin2xsinxcosx

=1sinxcosx

=2sin(2x)

=2csc(2x)

Dec 7, 2017

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

Explanation:

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

Then we bring under a common denominator:
=1tan(x)+tan(x)tan(x)tan(x)=1+tan2(x)tan(x)

Now we can use the tan2(x)+1=sec2(x) identity:
=sec2(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:
=(ca)2ba=c2a2ab=c2ab=cacb

Now if we convert back to trigonometric functions, we get:
cacb=sec(x)csc(x)

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