Answer:
Options A and D.
Explanation:
To check the identities we have to check each option given.
A. tan(x - π) = tanx
We take left hand side (L.H.S.) of the equation.
tan(π - x) = tanx [ since we know tan(π ± x) = tanx ]
So L.H.S. = R.H.S.
It's an identity.
B. sin(x + y) + sin(x - y) = 2cosx siny
We take L.H.S. first
sin(x + y) + sin(x - y)
= sinx coxy + siny cosx + sinx cosy - cosx siny
= 2sinx cosy ≠ R.H.S.
Option B is not an identity.
C. cos(x + y) - cos(x - y) = 2cosx cosy
We take L.H.S. of the equation.
cos(x + y) - cos(x - y) = cosx cosy - sinx siny - (cosx cosy + sinx siny)
= -2sinx siny ≠ R.H.S.
Therefore, option C is not an identity.
D. cos(x + y) + cos(x - y) = 2cosx cosy
We take L.H.S. of the equation.
cos(x + y) + cos(x - y) = cosx cosy - sinx siny + cosx cosy + sinx siny
= 2cosx cosy = R.H.S.
Therefore, option D is an identity.
Options A and D are the identities.