Question

For each of the following determine whether or not it is an identity and prove your result.

a. cos(x)sec(x)-sin^2(x)=cos^2(x)

b. tan(x+(pi/4))= (tan(x)+1)/(1-tan(x))

c. (cos(x+y))/(cos(x-y))= (1-tan(x)tan(y))/(1+tan(x)tan(y))

d. (tan(x)+sin(x))/(1+cos(x))=tan(x)

e. (sin(x-y))/(sin(x)cos(y))=1-cot(x)tan(y)

f. sin(x)sin(y)=(1/2)[cos(x-y)-cos(x+y)]

Answer 1

a
1 - s^2 = c^2
yes
s^2+c^2 = 1

b
t(x+pi/4) = (t x + 1)/(1-t x)
yes

c
t x t y = s x /c x * s y/c y
so on the right we have
[1 - (s x s y / c x c y) ] / [ 1 + ( s x s y /c x c y) ]

[c x c y - s x s y] / [c x c y + s x s y]
which is indeed
c(x+y)/c(x-y)
so yes

d I am getting bored. I think you cn see the plan.