Trig Functions?

Question

Out of 46 problems, the following 7 are ones i could not figure out. Anyone able to help me through the answers in these. It'd be really appreaciated.


1. (sec x - tan x)^2 = 1-sinx/1+sinx

2. (cot^2x+1)(sin^2x+1) = cot^2x+2

3. cosθ + tanθsinθ = secθ

4. cos^3x + cosx sin^2x = cosx

5. cos^4x - sin^4x = 1-2sin^2x

6. 1-tan^4x = 2sec^2x - sec^4x

7. tan x + cot x = csc x * secx


I've tried to answer these but i am not getting it. Tahnks for the help!

Answer 1

#1
(sec x - tan x)^2
=(secx)^2 -2secxtanx + (tanx)^2
= 1/(cosx)^2 - 2sinx/(cosx)^2 + (sinx)^2/(cosx)^2
= [ 1 - 2sinx + (sinx)^2 ] / (cosx)^2
= (1 - sinx)^2 / (cosx)^2
= (1 - sinx)^2 / [ 1 - (sinx)^2 ]
= (1 - sinx)^2 / [ (1 - sinx)(1 + sinx) ]
= (1 - sinx) / (1 + sinx)

#3
cosθ + tanθsinθ
= cosθ + (sinθ/cosθ)sinθ
= cosθ + (sinθ)^2/cosθ
= (cosθ)^2/cosθ +(sinθ)^2/cosθ
= [ (cosθ)^2 + (sinθ)^2 ] / cosθ
= 1 / cosθ
= secθ

#7
tanx + cotx
= sinx/cosx + cosx/sinx
= (sinx)^2/(sinxcosx) + (cosx)^2/(sinxcosx)
= [ (sinx)^2 + (cosx)^2 ] / (sinxcosx)
= 1 / (sinxcosx)
= (1/sinx)(1/cosx)
= cscx * secx