Evaluate each expression without using a calculator.
cos (2 arctan (4/3))

Verified step by step guidance

1. Recognize that the expression involves a double angle cosine function with an angle defined as \(\theta = \arctan\left(\frac{4}{3}\right)\). So the expression is \(\cos(2\theta)\) where \(\theta = \arctan\left(\frac{4}{3}\right)\).
2. Recall the double angle identity for cosine: \(\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}\). This identity is useful because we know \(\tan(\theta)\) from the problem.
3. Substitute \(\tan(\theta) = \frac{4}{3}\) into the double angle formula: \(\cos(2\theta) = \frac{1 - \left(\frac{4}{3}\right)^2}{1 + \left(\frac{4}{3}\right)^2}\).
4. Simplify the numerator and denominator separately by squaring \(\frac{4}{3}\) and then performing the subtraction and addition inside the fraction.
5. After simplification, write the resulting fraction as the value of \(\cos(2\arctan(4/3))\). This completes the evaluation without using a calculator.

Key Concepts

Inverse Trigonometric Functions (Arctan)

The arctan function, or inverse tangent, returns an angle whose tangent is a given number. For example, arctan(4/3) gives an angle θ such that tan(θ) = 4/3. Understanding this allows us to express trigonometric expressions involving arctan in terms of right triangle ratios.

Double-Angle Formula for Cosine

The double-angle formula for cosine states that cos(2θ) = cos²θ - sin²θ, or equivalently cos(2θ) = 1 - 2sin²θ or 2cos²θ - 1. This formula helps simplify expressions involving twice an angle, such as cos(2 arctan(x)), by relating it to sine and cosine of the original angle.

Right Triangle Trigonometry and Ratio Conversion

By interpreting arctan(4/3) as an angle in a right triangle with opposite side 4 and adjacent side 3, we can find the hypotenuse and then determine sine and cosine values. This geometric approach allows us to rewrite trigonometric expressions in exact fractional form without a calculator.