cosx cosy Formula | cosx cosy Identity

Note that cosx cosy is the product of two cosine functions cosx and cosy. The formula of the product cosx cosy is given as follows:

cosx cosy = $\dfrac{\cos(x+y)+\cos(x-y)}{2}$

Proof of cosx cosy Formula

Let us now prove the cosx cosy formula

cosx cosy = $\dfrac{\cos(x+y)+\cos(x-y)}{2}$

Proof:

We know that

cos(x+y) = cosx cosy – sinx siny …(I)

cos(x-y) = cosx cosy + sinx siny …(II)

Adding (I) and (II), we get that

cos(x+y) + cos(x-y) = (cosx cosy – sinx siny) + (cosx cosy + sinx siny)

⇒ cos(x+y) + cos(x-y) = cosx cosy – sinx siny + cosx cosy + sinx siny

⇒ cos(x+y) + cos(x-y) = 2 cosx cosy

⇒ cosx cosy = $\frac{1}{2}$ [cos(x+y) + cos(x-y)]

So cosx cosy is equal to 1/2 [cos(x+y) + cos(x-y)].

cosx cosy Formula:

$\cos x \cos y = \dfrac{\cos(x+y)+\cos(x-y)}{2}$

ALSO READ:

sinx cosy Formula

sinx siny Formula

cosx siny Formula

Application of cosx cosy Formula

Question 1: Find the value of cos45 cos15.

Answer:

By the above formula,

cos45 cos15 = $\dfrac{\cos(45+15)+\cos(45-15)}{2}$

= $\dfrac{\cos 60+\cos 30}{2}$

= $\dfrac{\frac{1}{2}+\frac{\sqrt{3}}{2}}{2}$

= $\dfrac{\frac{1+\sqrt{3}}{2}}{2}$

= $\dfrac{1+\sqrt{3}}{4}$

So the value of cos45 cos15 is equal to (1+√3)/4 and this is obtained by applying the cosx cosy formula.

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