Write each function as an expression involving functions of θ or x alone. See Example 2.
tan (π/4 + x)

Verified step by step guidance

Recall the tangent addition formula: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\).

Identify \(a = \frac{\pi}{4}\) and \(b = x\) in the expression \(\tan\left(\frac{\pi}{4} + x\right)\).

Substitute \(a\) and \(b\) into the formula: \(\tan\left(\frac{\pi}{4} + x\right) = \frac{\tan\frac{\pi}{4} + \tan x}{1 - \tan\frac{\pi}{4} \tan x}\).

Recall that \(\tan\frac{\pi}{4} = 1\), so simplify the expression to \(\frac{1 + \tan x}{1 - 1 \cdot \tan x} = \frac{1 + \tan x}{1 - \tan x}\).

Thus, \(\tan\left(\frac{\pi}{4} + x\right)\) is expressed solely in terms of \(\tan x\) as \(\frac{1 + \tan x}{1 - \tan x}\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angle Sum Identity for Tangent

The angle sum identity for tangent states that tan(a + b) = (tan a + tan b) / (1 - tan a * tan b). This formula allows expressing the tangent of a sum of two angles in terms of the tangents of the individual angles, which is essential for rewriting tan(π/4 + x).

Value of Tangent at Special Angles

Knowing the exact values of tangent at special angles like π/4 is crucial. Since tan(π/4) = 1, this simplifies the expression when applying the angle sum identity, making it easier to rewrite tan(π/4 + x) in terms of tan x.

Simplification of Rational Expressions

After applying the angle sum identity, the resulting expression often involves a rational function of tan x. Understanding how to simplify such fractions by factoring or combining terms is important to write the function in a clear, simplified form.

Recommended video: