Can someone help me with these trig problems.?

Question

I have 3 left and I can not get them.

(1-cosx)/(1+cos x)= (cot x - csc x)^2

((1)/(tan a - sec a))+((1)/(tan a + sec a))= -2 tan a

(csc x +cot x)/(tan x +sin x) = (cot x*csc x)

Thank you in advance for your help.

Answer 1

1) Multiply the left by (1-cosx)/(1-cosx). This gives (1-cosx)^2/(1-cox^2x)=(1-cosx)^2/sin^2(x)
Square the right, replace the functions with sin and cos and you get cos^2(x)/sin^2(x)-2cosx/sin^2(x)+1/sin^2(x). Rewrite as a single fraction over the denominator sin^2(x) and simplify to (1-cosx)^2/sin^2(x)
2) Combine fractions on the left and you get
2tanx/(tan^2(x)-sec^2(x)) By using the identity
tan^2(x)+1=sec^2(x) we see the denominator is -1, proving the identity.
3) Rewrite all functions on the left using sin and cos.
Multiply the numerartor and denominator by (sinx)(cosx) thus simplifying the fraction. We then can factor getting
[(cosx)(1+cosx)]/[sin^2(x)(1+cosx)]=cosx/sin^2(x) which is equivalent to the riight hand side.