Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

Question

How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B) $$But I am not sure how to proceed.

Answer 1

\begin{eqnarray} \sin(A+B)\sin(A-B) &=& (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B)\\ &=& \sin^2 A \cos^2 B -\sin^2 B \cos^2 A\\ &=& \sin^2 A \cos^2 B -\sin^2 B (1-\sin^2 A)\\ &=& \sin^2 A (\cos^2 B + \sin^2 A) - \sin^2 B\\ &=& \sin^2 A - \sin^2 B \end{eqnarray}

Answer 2

$$ \begin{align*}\sin (A+B)\cdot\sin (A-B)&=\frac{\cos(2B)-\cos (2A)}{2}\\&=\frac{(1-2\sin^2B)-(1-2\sin^2A)}{2}\\&=\sin^2A-\sin^2B\end{align*}$$

Here i have used $$\sin x\cdot\sin y=\frac{\cos(x-y)-\cos(x+y)}{2}$$

Answer 3

Hint: $(a+b)(a-b)=a^2-b^2$.
Then use $\sin^2\theta+\cos^2\theta=1$

Answer 4

use this formula

$$ 2Sin(A+B)Sin(A-B)=Cos2B-Cos2A $$

it will like this

$$ 1/2 * (Cos2B-Cos2A) $$ $$ \frac {(1-2Sin^2B)-(1-2sin^2A)}{2} $$

it will give answer

Answer 5

Open the bracket of what you got and substitute every cos^2 with 1-sin^2 and open bracket again. Thats it...

Answer 6

Your question involves the basic algebra identity which says, $(a + b)(a - b) = a^2 - b^2 $. For targeting your question, it is easy to assume $ a = \sin A\cos B $ and $b = \cos A \sin B$. The process becomes easy now.

$$\begin{align}(a + b)(a - b)& =& a^2 - b^2\\ & = & (\sin A \cos B)^2 - (\cos A \sin B)^2\\ & = & \sin^2A\cos^2 B - \cos^2A\sin^2B \\ & = & \sin^2A(1 - \sin^2 B) - \cos^2 A\sin^2B \end{align} $$Proceed.