1 tan 1 sin(2 ) Prove: . 1 tan cos(2 ) Start with: 1 tan 1 tan sin cos sin

Trigonometry, Mckeague, 6th edition
Homework Proof
5.3 #53
Prove:
1 tan x
1 tan x
Start with:
1 tan x
1 tan x
sin x
1
cos x
sin x
1
cos x
cos x sin x
cos x
cos x sin x
cos x sin x
1 sin(2 x)
.
cos(2 x)
cos x sin x
cos x cos x
cos x
sin x
cos x cos x
cos x
cos x sin x
cos x sin x
cos x
cos x sin x
cos x
Now this is a problem. I don't have any identities to use here.
I need squares to get some possiblities for identities.
One way to make squres is multiple by the conjugate.
cos x sin x
cos x sin x
cos x sin x
cos x sin x
cos 2 x 2sin x cos x sin 2 x
cos 2 x sin 2 x
1 2sin x cos x
.
cos(2 x)
Therefore,
1 tan x
1 tan x
The "fancy one" keeps the expression equivalent still.
The result of FOILing. Note all the squares and identities.
1 sin(2 x)
.
cos(2 x)