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Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains

Statement

If $\cos u \cos v \cos (u + v) \neq 0$ , then $\tan (u + v) = \frac{\tan u + \tan v}{1 - \tan u \tan v}, \sec (u + v) = \frac{\sec u \sec v}{1 - \tan u \tan v};$ if $\cos u \cos v \cos (u - v) \neq 0$ , then $\tan (u - v) = \frac{\tan u - \tan v}{1 + \tan u \tan v}, \sec (u - v) = \frac{\sec u \sec v}{1 + \tan u \tan v} .$ If $\sin u \sin v \sin (u + v) \neq 0$ , then $\cot (u + v) = \frac{\cot u \cot v - 1}{\cot u + \cot v}, \csc (u + v) = \frac{\csc u \csc v}{\cot u + \cot v};$ if $\sin u \sin v \sin (u - v) \neq 0$ , then $\cot (u - v) = \frac{\cot u \cot v + 1}{\cot v - \cot u}, \csc (u - v) = \frac{\csc u \csc v}{\cot v - \cot u} .$ The conventions and prerequisite facts used below are recorded in Tangent, cotangent, secant, and cosecant on their exact natural domains, The addition formulas for sine and cosine, The subtraction formulas for sine and cosine.

Facts & Assumptions

Given: Reals $u, v$ satisfying the relevant displayed nonvanishing hypothesis.

[L1]

Tangent, cotangent, secant, and cosecant on their exact natural domains defines $\tan t = \sin t / \cos t$ , $\cot t = \cos t / \sin t$ , $\sec t = 1 / \cos t$ , and $\csc t = 1 / \sin t$ on their natural domains.

[L2]

The addition formulas for sine and cosine gives the sine and cosine formulas for $u + v$ .

[L3]

The subtraction formulas for sine and cosine gives $\sin (u - v) = \sin u \cos v - \cos u \sin v$ and $\cos (u - v) = \cos u \cos v + \sin u \sin v$ .

Proof

technique · direct

1.1

Under the cosine nonvanishing hypothesis, divide the two formulas in [L2] by $\cos u \cos v$ . Their quotient gives the tangent addition formula, and their cosine formula gives $\cos (u + v) = \cos u \cos v (1 - \tan u \tan v)$ . Taking reciprocals by [L1] gives the secant addition formula.

L1L2

1.2

Under the sine nonvanishing hypothesis, divide the formulas in [L2] by $\sin u \sin v$ . Their quotient gives the cotangent addition formula, and their sine formula gives $\sin (u + v) = \sin u \sin v (\cot u + \cot v)$ . Taking reciprocals by [L1] gives the cosecant addition formula.

L1L2

2.1

The two formulas in [L3], divided by the same nonzero products as in steps 1.1 and 1.2, give respectively the tangent--secant and cotangent--cosecant subtraction formulas. Every denominator used is nonzero by the hypothesis displayed next to that formula.

L1L3∎

Depends on

Used by

u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions Counterexample
Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)