Question

Question #f2ee0

Answer 1

Convert to the pythagorean identity `cos^2(x)=1-sin^2(x)`

Explanation:

1. `sec(x)-tan(x)=1/cos(x)-sin(x)/cos(x)` (converted right side to `sin` and `cos`)
2. `cos(x)/(1+sin(x)) = (1 - sin(x))/cos(x)` (now you have this)
3. `cos^2(x)=1-sin^2(x)` (cross multiply)
4. The above is the pythagorean identity for sin/cos.

Answer 2

See below.

Explanation:

Formatted question: Prove: `cosx/(1+sinx)=secx-tanx`

We can start from either the left-hand side (LHS) or right-hand side (RHS) and work toward the other side or work from both sides independently to an equal point. In this proof, we will prove the given identity from the left-hand side to the right-hand side:

`LHS=cosx/(1+sinx)`

We can multiply the numerator and the denominator by `(1-sinx)`:
`=(cosx(1-sinx))/((1+sinx)(1-sinx))`
`=(cosx(1-sinx))/(1-sin^2x)`

Consider the Pythagorean Identity: `sin^2x+cos^2x=1`, which can be rearranged to `cos^2x=1-sin^2x`, which is more applicable in this problem.

We can substitute `cos^2x=1-sin^2x` into the previous step:
`LHS=(cosx(1-sinx))/cos^2x`
`=(1-sinx)/cosx`
`=1/cosx-sinx/cosx`
`=secx-tanx`
`=RHS`

`thereforecosx/(1+sinx)=secx-tanx`