NIMCET 2021 — Mathematics PYQ

We use the half-angle identity for cosine: ${\cos }^2\frac{\theta }{2}=\frac{1+\cos \theta }{2}$.

Step 1: Substitute the identity into the given equation.

Multiplying the entire equation by $2$ to clear the denominators:

Step 2: Apply the Projection Rule.

The Projection Rule in a triangle states:

Substituting this into our equation:

Step 3: Simplify the expression.

Conclusion:

The condition $a+c=2b$ is the standard definition for an Arithmetic Progression (AP).

Therefore, the sides $a,b,c$ are in AP.