Proving trigonometric identities

Question

Proof #1: 2

proof #2: 9

proof#3: 3

a.) -sin x -cos x cot x =-csc x

b.) 1+ tan x/ 1 + cot x = tan x

c.) sin x/1 + cos x = 1 - cos x/sin x

so I am supposed to prove the above numbers 2,9 and 3 and the can be used in any order so b for proof 3 and so on. The numbers are trig identities that need to be proved in the following equations

Answer 1

a) -sinx -cosxcotx = -cscx

-sinx - cosx(cosx/sinx) = -1/sinx

multiply by sinx

-sin^2x - cos^2x = -1

multiply by -1

sin^2x + cos^2 = 1 which is a trig identity

b) (1+tanx)/(1+cotx) = tanx

if you don't know what to do, or just to see better what's going on, convert everything into sines and cosines. they're easier to work with, generally

(1+ sinx/cosx)/(1+cosx/sinx) = sinx/cosx

= (cosx+sinx)/cosx)/(sinx+cosx)/sinx) = sinx/cosx

cross multiply

sinx+cosx = cosx + sinx

c) sinx/(1+cosx) = (1-cosx)/sinx

cross multiply to get

sinx(sinx) = (1-cosx)(1+cosx)

sin^2x = 1-cos^2x. add cos^2 to both sides

sin^2x + cos^2 = 1 is a trigonometric identity, trigonometry's version of the Pythagorean Theorem.

sinx = opposite side over hypotenuse = o/h

sin^2x = o^2/h^2

cosx = adjacent side over hypotenuse = a/h

cos^2x = a^2/h^2

sin^2x + cos^2x = o^2/h^2 + a^2/h^2 = (o^2+a^2)/h^2 = h^2/h^2 = 1