Which of the following are identities? check all that apply.

a. tan(x-pi/4)=tanx-1
b. 1-tanxtany=(sin(x+y))/sinxsiny
c. tanx-tany=sin(x-y)/cosxcosy
d. cos(x+pi/6)=sin(x-pi/3)

C

Explanation:

Test A.
The left side is
tan(x - π/4) = [tan(x) - tan(π/4)]/[1 + tan(x)*tan(π/4)]
                   = [tan(x) - 1]/[1 - tan(x)]
                  = -1
This is not equal to the right side.
Statement A is not an identity.

Test B.
The right side is
sin(x+y)/(sinx siny) = [sin(x)cos(y) + cos(x)sin(y)]/[sin(x)sin(y)]
                               = cot(y) + cot(x) = 1/tan(y) + 1/tan(x)
                               = [tanx + tany]/[tan(x)tan(y)]
This is not equal to the left side.
Statement B is not an identity.

Test C.
The right side is
[sin(x)cos(y) - cos(x)sin(y)]/[cos(x)cos(y)]
= sin(x)/cos(x) - sin(y)/cos(y)
= tan(x) - tan(y)
Ths equals the left side.
Statement C is an identity.

Test D.
The left side is
cos(x)cos(π/6) - sin(x)sin(π/6)
= (√3/2)cos(x) - (1/2)sin(x).

The right side is
sin(x)cos(π/3) - cos(x)sin(π/3)
= (1/2)sin(x) - (√3/2)cos(x)

The two sides are not equal.
Statement D is not an identity.