Question

If \[2\cos A.\cos B = \cos x + \cos y\] then \[x + y\]=?

Answer 1

Hint: We need to know trigonometric formulas of sum and difference. So to find the value of \[x + y\] we will change the left side of the equation with the formula. And then we will compare the two sides and get the value of \[x + y\].

Complete step by step answer:
Given that,
\[2\cos A.\cos B = \cos x + \cos y\]
Now as we know that, \[2\cos A.\cos B = \cos (A + B) + \cos (A - B)\]
So we will replace the LHS,
\[ \Rightarrow \cos (A + B) + \cos (A - B) = \cos x + \cos y\]
Now both the sides have two cos functions with plus sign in between so we can compare the terms directly.
\[x = A + B\& y = A - B\]
Now we have values of both, x and y thus we can find the value of \[x + y\]
\[x + y = A + B + A - B\]
\[x + y = 2A\]
Thus this is the exact answer we have to find.

Note:
Note that, in the trigonometry question above it was possible for us to equate or compare the sides because both sides have the same functions and same sign. Otherwise we cannot compare them. We have a double angle and triple angle formula as well. Sometimes those also can be used to solve the problems. Also note that, we can also find any value that involves x and y term with any operation sign now because we have value of both x and y.