Question

How do you simplify 1+cos(x)/1+csc(x)?

Answer 1

`(1+cosx)/(1+cscx) = (2sinx+sin2x)/(2(sinx+1))`

Explanation:

Start by splitting the numerator and denominator and simplifying.

`1 + cscx = 1 + 1/sinx`
`=(sinx +1)/sinx`

Now perform the division (multiply the numerator by the reciprocal denominator):

`(1+cosx)*(sinx)/(sinx +1) = (sinx(1+cosx))/(sinx +1)`

This fraction becomes: `(sinx + sinx cosx)/(sinx +1)`

By the double angle formula for `sin`, `sinxcosx=1/2sin2x`.

Bringing out the factor of `1/2` in the numerator, the fraction now becomes: `(1/2(2sinx +sin2x))/(sinx +1)`

Simplify to find the fraction: `(2sinx + sin2x)/(2(sinx+1))`.