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

Statement

If cos⁡ucos⁡vcos⁡(u+v)≠0\cos u\cos v\cos(u+v)\ne0, then tan⁡(u+v)=tan⁡u+tan⁡v1−tan⁡utan⁡v,sec⁡(u+v)=sec⁡usec⁡v1−tan⁡utan⁡v;\tan(u+v)=\frac{\tan u+\tan v}{1-\tan u\tan v},\qquad \sec(u+v)=\frac{\sec u\sec v}{1-\tan u\tan v}; if cos⁡ucos⁡vcos⁡(u−v)≠0\cos u\cos v\cos(u-v)\ne0, then tan⁡(u−v)=tan⁡u−tan⁡v1+tan⁡utan⁡v,sec⁡(u−v)=sec⁡usec⁡v1+tan⁡utan⁡v.\tan(u-v)=\frac{\tan u-\tan v}{1+\tan u\tan v},\qquad \sec(u-v)=\frac{\sec u\sec v}{1+\tan u\tan v}. If sin⁡usin⁡vsin⁡(u+v)≠0\sin u\sin v\sin(u+v)\ne0, then cot⁡(u+v)=cot⁡ucot⁡v−1cot⁡u+cot⁡v,csc⁡(u+v)=csc⁡ucsc⁡vcot⁡u+cot⁡v;\cot(u+v)=\frac{\cot u\cot v-1}{\cot u+\cot v},\qquad \csc(u+v)=\frac{\csc u\csc v}{\cot u+\cot v}; if sin⁡usin⁡vsin⁡(u−v)≠0\sin u\sin v\sin(u-v)\ne0, then cot⁡(u−v)=cot⁡ucot⁡v+1cot⁡v−cot⁡u,csc⁡(u−v)=csc⁡ucsc⁡vcot⁡v−cot⁡u.\cot(u-v)=\frac{\cot u\cot v+1}{\cot v-\cot u},\qquad \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,vu,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\tan t=\sin t/\cos t, cot⁡t=cos⁡t/sin⁡t\cot t=\cos t/\sin t, sec⁡t=1/cos⁡t\sec t=1/\cos t, and csc⁡t=1/sin⁡t\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+vu+v.

[L3]

The subtraction formulas for sine and cosine gives sin⁡(u−v)=sin⁡ucos⁡v−cos⁡usin⁡v\sin(u-v)=\sin u\cos v-\cos u\sin v and cos⁡(u−v)=cos⁡ucos⁡v+sin⁡usin⁡v\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⁡ucos⁡v\cos u\cos v. Their quotient gives the tangent addition formula, and their cosine formula gives cos⁡(u+v)=cos⁡ucos⁡v(1−tan⁡utan⁡v)\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⁡usin⁡v\sin u\sin v. Their quotient gives the cotangent addition formula, and their sine formula gives sin⁡(u+v)=sin⁡usin⁡v(cot⁡u+cot⁡v)\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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources