Question

Show that show that for all reals `a` and `x` we ​​have?

`sin(a+x)*cos(a-x)+cos(a+x)*sin(a-x)=sin2a`

Answer 1

See explanation...

Explanation:

Use the angle sum formula for `sin`...

`sin(A+B) = sin A cos B + cos A sin B`

with `A=a+x` and `B=a-x` as follows:

`sin 2a = sin ((a+x)+(a-x))`

`color(white)(sin 2a) = sin(a+x) cos(a-x)+cos(a+x) sin(a-x)`

Answer 2

Its proved

Explanation:

Let a+x= b
a-x= c

`sin(a+x)cos(a-x) + cos(a+x)sin(a-x)`
=`sinb*cosc + cosb*sinc`
We know , `sinx*cosy + cosx*siny =sin(x+y)`

So,`sinb*cosc + cosb*sinc = sin(b+c)`
Put the values of b and c
= sin(a+x+a-x)
= sin(2a)

Hope it helps you