Tons of trig questions

[shortnsweet]

I need to prove:
1. secx/(secx-tanx)=sec^2x+secxtanx
2. sin^3xcos^2x=sin^3x-sin^5x
3. sec^2x+csc^2x=sec^2xcsc^2x
4. (1-3cosx-4cos^2x)/sin^2x=(tanx-4)/(tanx+2)
5. csc^5x-cot^6x=1+3csc^2xcot^2x
6. (1+sinx+cosx)/(1+sinx-cosx)=(1+cosx)/sinx
I've always had A's in math but if i'm having this many problems should I get tutoring?

[krtxmrtz]

Just do some manipulation according to the definition:

sec x = 1 / cos x

For example, the first question:

sec x / (sec x - tan x) = (1 / cos x) / (1 / cos x - sin x / cos x) = 1 / (1 - sin x) = (1 + sin x) / [(1 + sin x)(1 - sin x)] = (1 + sin x) / (1 - sin² x) = (1 + sin x) / cos² x = 1 / cos² x + (1 / cos x)(sin x / cos x) = sec² x + sec x tan x

[krtxmrtz]

As for number 2:

sin³x cos²x = sin³x (1 - sin²x) =sin³x - sin⁵x

I think this should get you going.