Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. \(\arctan x=\frac{\arcsin x}{\arccos x}\)

Step-by-Step Solution

Answer

The statement \(\arctan x=\frac{\arcsin x}{\arccos x}\) is indeed true.

Step 1: Recall definitions

Here, it's important to remember the foundational properties of arccos, arcsin, and arctan functions.

Step 2: Convert to Cotangent

Next, to simplify the comparison, we convert \(\arctan x\) into \(\frac{1}{\tan x}\), and \(\frac{\arcsin x}{\arccos x}\) into \(\frac{1}{\cos x}\) and \(\frac{1}{\sin x}\) respectively.

Step 3: Use Trigonometric Tautology

The relation \(\sin(x)^2 + \cos(x)^2 = 1\) is well known, so we can safely combine our expression into \(\frac{1}{\sin x}\) and \(\frac{cos x}{sin x}\) which results in cotangent.

Step 4: Examine the Results

In the end, we will see that the left side \(\frac{1}{\tan x}\) is indeed equal to \(\frac{cos x}{sin x}\) which stands for the right side.