d) cos(x + y) cos(x − y) = cos2 x + cos2 y − 1
To solve the equation cos(x + y) cos(x − y) = cos2 x + cos2 y − 1, we will begin by expanding both sides of the equation.
Using the identity cos(A + B) = cos A cos B − sin A sin B, we can rewrite the left side of the equation as follows:
cos(x + y) cos(x − y) = (cos x cos y − sin x sin y) (cos x cos y + sin x sin y)
= cos2 x cos2 y − sin2 x sin2 y
= cos2 x (1 - sin2 y) - (1 - cos2 x) sin2 y
= cos2 x - cos2 x sin2 y - sin2 y + cos2 x sin2 y
= cos2 x - sin2 y
Now, we can simplify the right side of the equation:
cos2 x + cos2 y − 1 = cos2 x + (1 - sin2 y) - 1
= cos2 x + 1 - sin2 y - 1
= cos2 x - sin2 y
Thus, we have shown that cos(x + y) cos(x − y) = cos2 x + cos2 y − 1 is true.