Let D and E be the unions of open intervals defined as...

Let $D$ and $E$ be the unions of open intervals defined as follows. $D=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+1) \pi}{2}\right)$ and $E=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-3) \pi}{2}, \frac{(4 k-1) \pi}{2}\right)$ Show that $\sin x=\left\{\begin{array}{rl}\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in D, \\ -\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in E,\end{array} \quad \cos x=\left\{\begin{array}{c}\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in D, \\ -\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in E .\end{array}\right.\right.$

Let $D$ and $E$ be the unions of open intervals defined as follows.
$D=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+1) \pi}{2}\right)$ and $E=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-3) \pi}{2}, \frac{(4 k-1) \pi}{2}\right)$
Show that
$\sin x=\left\{\begin{array}{rl}\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in D, \\ -\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in E,\end{array} \quad \cos x=\left\{\begin{array}{c}\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in D, \\ -\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in E .\end{array}\right.\right.$

Key Concepts

Piecewise-defined Functions in Trigonometry

Piecewise-defined functions allow for the accurate representation of trigonometric functions that change sign depending on the domain. When expressing sine and cosine in terms of tan x, different formulas must be used over specific intervals to maintain consistency with the original functions’ behaviors.

Quadrant Analysis and Sign Determination

The signs of sine and cosine depend on the quadrant in which the angle resides. This analysis is crucial when rewriting trigonometric functions because choosing the correct sign (positive or negative) for expressions like sqrt(1 + tan²x) ensures that the resulting formulas accurately reflect the behavior of these functions over different intervals.

Pythagorean Identity

The Pythagorean Identity, sin²x + cos²x = 1, is fundamental in trigonometry and allows one to express one trigonometric function in terms of another. In this context, dividing by cos²x gives 1 + tan²x = 1/cos²x, which forms the basis for writing cos x and sin x in terms of tan x.

Relationship Between Sine, Cosine, and Tangent

Tangent is defined as the ratio of sine to cosine, tan x = sin x/cos x. By using this definition together with the Pythagorean Identity, one can derive expressions for sine and cosine solely in terms of tan x, after carefully handling the positive and negative square root values.

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