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:**

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

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## 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.

## FAQs

**Q1: What is the formula of cosx cosy?**

**Answer:** The formula of cosx cosy is given by cosx cosy = 1/2 [cos(x+y)+cos(x-y)].

**Q2: What is the formula of cosa cosb?**

**Answer:** The formula of cosa cosb is given by cosa cosb = 1/2 [cos(a+b)+cos(a-b)].

**Q3: What is the formula of 2cosx cosy?**

**Answer:** The formula of 2cosx cosy is given by 2cosx cosy = cos(x+y)+cos(x-y).

**Q4: What is the formula of 2cosa cosb?**

**Answer:** The formula of 2cosa cosb is given by 2cosa cosb = cos(a+b)+cos(a-b).