Regarding real rate for perpetuity

It is not different. The formula for the value of the share is derived by discounting the dividend at the real rate.

But there was growth. isn't it similar to inflation? and if the rate of required return is the real rate shouldn't the default cost of capital also be real cost of capital?

Yes, arithmetically dividend growth is the same as inflation. I do not know what you mean by 'default cost of capital'. When appraising projects, the we discount the nominal/actual cash flows at the actual cost of capital, or alternatively (and there is no choice if it is a perpetuity) we discount the current price flows at the real cost of capital (the cost of capital with the inflation removed). When calculating the market value of a share, it is the PV of the dividends discounted at the shareholders actual required rate of return. If there is dividend growth in perpetuity, we discount the current dividend at the real required rate of return (the actual required return with inflation removed). The formula is doing exactly this.

Sir if that is the case then will we need to convert to actual rate after wacc calculation since it is the real rate wacc

Sir forgive me if I wasn't clear enough. What I am trying to say is is it possible to calculate the present value of of cashflows that are in perpetuity with inflation with the same formula we use to calculate the market value of share? replacing growth with inflation and the cost of capital would be the actual cost of capital? is this possible?

Yes, using the same formula will give exactly the same result and is fine :-)

So in the exam if actual cost of capital is given and inflation is also given can I use formula (with growth)

Yes you can :-)