Question

`(1+cot(x))/(sin(x)+cos(x))` the directions say to express the answer in terms of sin?

Answer 1

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

Explanation:

Given:

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

Substitute `cos(x)/sin(x) = cot(x)`:

`(1+cos(x)/sin(x))/(sin(x)+cos(x))`

Multiply by 1 in the form of `sin(x)/sin(x)`

`sin(x)/sin(x)(1+cos(x)/sin(x))/(sin(x)+cos(x))`

Multiply the numerators:

`1/sin(x)(sin(x)(1+cos(x)/sin(x)))/(sin(x)+cos(x))`

Distribute the sin(x):

`1/sin(x)(sin(x)+sin(x)cos(x)/sin(x))/(sin(x)+cos(x))`

Please observe that the sines cancel in the second term of the numerator:

`1/sin(x)(sin(x)+cos(x))/(sin(x)+cos(x))`

The second fraction becomes 1:

`1/sin(x)` this is in terms of the sine function.