The function sinx siny is the product of two sine functions sinx and siny. The formula or the identity of sinx siny is given as follows:

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

## Proof of sinx siny Formula

We will now prove the sinx siny formula

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

*Proof:*

We know that

cos(x+y) = cosx cosy – sinx siny **…(I)**

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

Subtracting **(I)** from **(II)**, that is, doing **(II)-(I)**, 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 sinx siny

⇒ sinx siny = $\frac{1}{2}$ [cos(x-y) – cos(x+y)]

Thus, sinx siny formula is given by sinx siny= 1/2 [cos(x-y) – cos(x+y)].

**sinx siny Formula:**

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

**ALSO READ:**

sinx cosy Formula | cosx siny Formula

cosx cosy Formula | 1-sin^{2}x Formula

Sin3x Formula in terms of sinx

Cos3x Formula in terms of cosx

Sin^{3}x Formula | Cos^{3}x Formula

## Application of sinx siny Formula

**Question 1: **Find the value of sin45 sin15.

*Answer:*

Applying the above formula with x=45 and y=15, we obtain that

sin45 sin15 = $\dfrac{\cos(45-15)-\cos(45+15)}{2}$

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

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

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

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

Hence the value of sin45 sin15 is equal to (√3-1)/4. We get this value by applying the sinx siny formula.

## FAQs

**Q1: What is the formula of sinx siny?**

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

**Q2: What is the formula of sina sinb?**

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

**Q3: What is the formula of 2sinx siny?**

Answer: The formula of 2sinx siny is given by 2sinx siny = cos(x-y)-cos(x+y).

**Q4: What is the formula of 2sina sinb?**

Answer: The formula of 2sina sinb is given by 2sina sinb = cos(a-b)-cos(a+b).