How do you prove that this trangle is isosceles?

Question

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

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Answer 1 · 2017-12-10T00:37:05

It is isosceles because $c=b$ .

.

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 $2sinb$ :

$cosc=cosb$

$c=b$