In Example 9, change sin to cos.

Data from Example 9

Find sin(tan⁻¹ x).
We know that tan⁻¹ x is another way of stating “the angle whose tangent is x.” Thus, let us draw a right triangle (as in Fig. 20.38) and label one of the acute angles as θ, the side opposite θ as x, and the side adjacent to θ as 1. In this way, we see that, by definition, tanθ = x/1 , or θ = tan⁻¹ x, which means θ is the desired angle. By the Pythagorean theorem, the hypotenuse of this triangle is

Now, we find that sinθ, which is the same as sin(tan⁻¹ x), is 

Thus

x + 1.