Question

How do you prove that this trangle is isosceles?

The letters a,b,c are for the angles of the triangle.

How to prove that:

if `sin(b + c) + sin(b - c) = sin2b` , than is the triangle isosceles

Answer 1

It is isosceles because `c=b`.

Explanation:

.

We use the following formula:

`sin(a+b)=sinacosb+cosasinb`

`sin(a-b)=sinacosb-cosasinb`

Therefore:

`sinbcosc+coscsinb+sinbcosc-coscsinb=sin2b`

`sinbcosc+cancelcolor(red)(coscsinb)+sinbcosc-cancelcolor(red)(coscsinb)=sin2b`

`2sinbcosc=sin2b`

But we have a double angle identity that says:

`sin2x=2sinxcosx`, therefore:

`2sinbcosc=2sinbcosb`

Divide both sides by `2sinb`:

`cosc=cosb`

`c=b`