The key point to solve the problem:

To prove a triangle isosceles our task is to show either any two angles equal or two sides equal.

Idea of cosine formula - Cos C =

The idea of sine Formula:

•

Given,

As it has sin terms involved so that sine formula can work, and cos C is also there so we might need cosine formula too.

Let’s apply sine formula keeping a target to prove any two sides equal.

Using sine formula we have –

∴ sin A = ak and sin B = bk

∴ cos C =

If we apply cosine formula, we will get an equation in terms of sides only that may give us any two sides equal.

Using, Cos C =

We have,

⇒ b² + a² – c² = a²

⇒ b² = c²

⇒ b = c

Hence 2 sides are equal.

∴ Δ ABC is isosceles. ….proved