Question

Question #545b9

Answer 1

`cos(x)(tan(x)+cot(x))=1/sin(x)`

Explanation:

We will use the definitions of the tangent and cotangent functions:

• `tan(x) = sin(x)/cos(x)`
• `cot(x) = cos(x)/sin(x)`

as well as the identity `sin^2(x)+cos^2(x) = 1`
Note that subtracting `sin^2(x)` from both sides produces the form we will use:

• `cos^2(x) = 1-sin^2(x)`

With those, we have

`cos(x)(tan(x)+cot(x)) = cos(x)(sin(x)/cos(x)+cos(x)/sin(x))`

`=cancel(cos(x))*sin(x)/cancel(cos(x))+cos(x)*cos(x)/sin(x)`

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

`=sin(x)+(1-sin^2(x))/sin(x)`

`=sin(x)+1/sin(x)-(sin(x)cancel(sin(x)))/cancel(sin(x))`

`=cancel(sin(x))+1/sin(x)-cancel(sin(x))`

`=1/sin(x)`