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

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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 \(\frac{1 + \tan x}{1 - \tan x}\).

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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).

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Sum and Difference of Tangent

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.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

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.

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