sin-1⁡x+cos1⁡x equals to

The correct option is π/2.

To understand why this is correct, let us look at the relationship between the inverse trigonometric functions. We want to find the value of the expression:

${\sin }^{-1}x+{\cos }^{-1}x$

Let us define a variable, say, $\theta$, as follows:

$\theta ={\sin }^{-1}x$

By the definition of the inverse sine function, this is equivalent to:

$x=\sin \theta$

Using the trigonometric co-function identity, we know that the sine of an angle is equal to the cosine of its complement:

$\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right)$

Substituting this back into our equation for $x$, we get:

$x=\cos \left(\frac{\pi }{2}-\theta \right)$

Taking the inverse cosine of both sides gives:

${\cos }^{-1}x=\frac{\pi }{2}-\theta$

Now, we substitute the definition of $\theta$ back into the equation:

${\cos }^{-1}x=\frac{\pi }{2}-{\sin }^{-1}x$

Finally, rearranging the terms by adding ${\sin }^{-1}x$ to both sides of the equation leads to:

${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$