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Chapter 5: Problem 60

Write the trigonometric expression as an algebraic expression. $$\cos (\arccos x-\arctan x)$$

Short Answer

Expert verified
The algebraic expression for \(\cos (\arccos x - \arctan x)\) is \(\frac{x}{\sqrt{1+x^2}} + \frac{x\sqrt{1-x^2}}{\sqrt{1+x^2}}\).

Step by step solution

01

Recall the trigonometric identity

We should first remember the trigonometric identity which is used to express cosine of a difference of two angles. It states that \(\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)\). Therefore the given expression \(\cos (\arccos x - \arctan x)\) can be written as \(\cos (\arccos x)\cos (\arctan x) +\sin (\arccos x) \sin (\arctan x)\).
02

Convert inverse trigonometric functions to algebraic form

Next, we convert the inverse trigonometric functions to their algebraic forms. Remember that \(\cos (\arccos x) = x\), and \(\cos (\arctan x) = \frac{1}{\sqrt{1+x^2}}\) from Pythagorean theorem. Also, \(\sin (\arccos x) = \sqrt{1-x^2}\), and \(\sin (\arctan x) = \frac{x}{\sqrt{1+x^2}}\). Thus the expression can be written in algebraic form as \(x \cdot \frac{1}{\sqrt{1+x^2}} + \sqrt{1-x^2} \cdot \frac{x}{\sqrt{1+x^2}}\).
03

Combine and simplify

The last step is to simplify the algebraic expression by multiplying and combining the terms, giving us \(\frac{x}{\sqrt{1+x^2}} + \frac{x\sqrt{1-x^2}}{\sqrt{1+x^2}}\).

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