Question

Cos (S+T) + cos (C-T)= 2 cos(s)cos(t)

Answer 1

cos(s)cost-sinssint+costcoss+sintsins=cosscost+costcoss=2cosscost

Answer 2

To prove the given trigonometric identity, we'll work with the left side of the equation and simplify it to match the right side:

Starting with the left side: cos(S+T) + cos(C-T)

Using the sum-to-product formula for cosine, we have: cos(S)cos(T) - sin(S)sin(T) + cos(C)cos(T) + sin(C)sin(T)

Rearranging the terms: (cos(S)cos(T) + cos(C)cos(T)) + (-sin(S)sin(T) + sin(C)sin(T))

Factoring out cos(T) and sin(T): cos(T)(cos(S) + cos(C)) + sin(T)(-sin(S) + sin(C))

Using the identity sin(-x) = -sin(x): cos(T)(cos(S) + cos(C)) - sin(T)(sin(S) - sin(C))

Now, let's focus on the right side of the equation: 2cos(S)cos(T)

We can see that the right side matches the simplified form we obtained from the left side.

Therefore, we have proven the given trigonometric identity: cos(S+T) + cos(C-T) = 2cos(S)cos(T)