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Chapter 5: Problem 49

Verify the identity. $$\tan \left(\sin ^{-1} \frac{x-1}{4}\right)=\frac{x-1}{\sqrt{16-(x-1)^{2}}}$$

Short Answer

Expert verified
By following these steps, we can verify that the given identity is true.

Step by step solution

01

Solve the inside of the inverse sine function

The inverse sine function, \( \sin ^{-1} \), takes a ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Thus, if \( \sin ^{-1} ( \frac{x-1}{4}) \), then the opposite side is \(x-1\), and the hypotenuse is 4.
02

Find the adjacent side of the triangle using the Pythagorean theorem

We can find the length of the adjacent side using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, the length of the adjacent side is \( \sqrt{4^2-(x-1)^2} = \sqrt{16-(x-1)^{2}} \).
03

Calculate the tangent of the angle

The definition of the tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Therefore, \( \tan ( \sin ^{-1} ( \frac{x-1}{4})) = \frac{x-1}{\sqrt{16-(x-1)^{2}}} \).

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