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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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mathsAdvanced TrigonometryTrigonometric Identities for compound angles

Compound Angles: tan(A+B) and tan(A-B)

In this page, expressing `tan(A + B)` and `tan(A-B)` in terms of `tan A` and `tan B` is explained.



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Proof for tan(A+B) and tan(A-B) using previous results and algebra of trigonometric functions.

`tan(A+B)`
`quad quad = (sin(A+B))/(cos(A+B))`
`quad quad = (sinA cosB + cosA sinB)/(cosA cosB - sinA sinB)`
    dividing numerator and denominator by `cosA cosB`
`quad quad = (tan A + tan B)/(1 - tan A tan B)`

`tan(A-B)`
`quad quad = tan(A+(-B))`
`quad quad = (tan A - tan B)/(1 + tan A tan B)`

`tan(A+B) = (tan A + tan B)/(1- tan A tan B)`
`tan(A-B) = (tan A - tan B)/(1+ tan A tan B)`

Proof for cot(A+B) and cot(A-B) using previous results and algebra of trigonometric functions.

`cot(A+B)`
`quad quad = (cos(A+B))/(sin(A+B))`
`quad quad = (cosA cosB - sinA sinB)/(sinA cosB + cosA sinB)`
    dividing numerator and denominator by `sinA sinB`
`quad quad = (cot A cot B -1)/(cot A + cot B)`

`cot(A-B)`
`quad quad = cot(A+(-B))`
`quad quad = (-cot A cot B -1)/(cot A - cot B)`

`cot(A+B) = (cot A cot B -1)/(cot A + cot B)`
`cot(A-B) = (-cot A cot B -1)/(cot A - cot B)`

Trigonometric Identities for compound Angles

`sin(A+B)=sinA cosB + cosA sinB`
`sin(A-B)=sinA cosB - cosA sinB`
`cos(A+B)=cosA cosB - sinA sinB`
`cos(A-B)=cosA cosB + sinA sinB`

`tan(A+B) = (tan A + tan B)/(1- tan A tan B)`
`tan(A-B) = (tan A - tan B)/(1+ tan A tan B)`

`cot(A+B) = (cot A cot B -1)/(cot A + cot B)`
`cot(A-B) = (-cot A cot B -1)/(cot A - cot B)`

                            
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