Question

How do you simplify the following?

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

2. `(tan x)(1-sin^2 x)`

Answer 1

1. `(1-sin^2 x)/(sin x + 1) = 1 - sin x` when `x != (3pi)/2 + 2kpi`

2. `(tan x)(1 - sin^2 x) = 1/2 sin 2x` when `x != kpi`

Explanation:

Example 1.

Use the difference of squares identity:

`a^2-b^2 = (a-b)(a+b)`

with `a = 1`, `b = sin x` as follows:

`(1-sin^2 x)/(sin x + 1) = ((1-sin x)color(red)(cancel(color(black)((1+sin x)))))/color(red)(cancel(color(black)((1+sin x)))) = 1 - sin x`

with exclusion `sin x != -1` (i.e. `x != (3pi)/2+2kpi`)

Example 2.

Use the following:

`sin^2 x + cos^2 x = 1` in the form `1 - sin^2 x = cos^2 x`

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

`sin 2x = 2 sin x cos x`

as follows:

`(tan x)(1 - sin^2 x) =(sin x)/(cos x)*cos^2 x = sin x cos x = 1/2 sin 2x`

with exclusion `cos x != 0`, i.e. `x != kpi`