If are in A.P., whose common difference is d, show that

Solution:

It is given that are in the form of A.P., and is the common difference,

We now need to prove that

When we cross multiply we obtain

It is known so utilizing this formula we obtain

Now consider the Left Hand Side

Left Hand Side

We now need to find the value of in terms of in order to simplify further.

As are in AP having common difference as

Therefore

On both sides taking sin

Now substitute the value of sin for each term in Left Hand Side

Left Hand Side

It is known that

Utilize this formula to obtain

Left Hand Side

On simplying

Known that

Left Hand Side = Right Hand Side

As a result, hence proved.