Prove that:

To prove:

We will consider the LHS, so

LHS =

Now let

So we need to need to find first x+y

As per our assumption,

Taking cos on both sides, we get

Taking square on both sides, we get

Now we know sin²x+cos²x=1 ⇒ cos²x=1-sin²x, so the above equation becomes,

Taking square root on both sides, we get

As per another assumption,

Taking cos on both sides, we get

Taking square on both sides, we get

Now we know sin²x+cos²x=1 ⇒ cos²x=1-sin²x, so the above equation becomes,

Taking square root on both sides, we get

Now we know,

Cos (x+y)=cos x cos y-sin x sin y

Now substituting values from equation (i), (ii), (iii) and (iv), we get

Now taking cos^-1 on both sides, we get

Now substituting the values of x and y, we get

Hence Proved