Find the principa

Let us understand what principal value of inverse trigonometric function is.

The principal value of an inverse trigonometric function, say, cos^-1 x for x > 0, is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose cosine is x. For this reason cos^-1 x is also denoted by arc cos x.

First, let us find principal value of . Let the principal value be x, such that

Note that, . So,

Range of principal value of sin^-1 is between and .

And, .

Hence, principal value of is .

Now, let is find principal value of . Let the principal value be y, such that

Note that, . So,

Since, range of principal value of cos^-1 is between 0 and π.

And, does not belong to [0, π].

So,

Hence, principal value of is .

Now, add the principal values.

Thus, principal value of is .