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Thought-Process to Discover Knowledge

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mathsAdvanced TrigonometryTrigonometric Identities for compound angles

More results for trigonometric values of Compound Angles

In this page, some additional results involving trigonometric function of compound angle is discussed.



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Consider `sin(A+B)sin(A-B)`
`quad = (sinA cosB + cosA sinB)`
`quad quad quad xx (sinA cosB - cosA sinB)`
`quad = sin^2 A cos^2B - cos^2A sin^2B`

substitute `cos^2 B = (1-sin^2 B)`
`quad = sin^2A - sin^2B xx(sin^2 A+cos^2 A)`
`quad = sin^2A - sin^2B`

substitute `sin^2A = (1-cos^2 A)`
`quad = cos^2B - cos^2A(cos^2B+sin^2B)`
`quad = cos^2 B - cos^2 A`

`sin(A+B)sin(A-B)`
`quad quad = sin^2 A - sin^2 B`
`quad quad = cos^2 B - cos^2 A`

Consider `cos(A+B)cos(A-B)`
`quad = (cosA cosB - sinA sinB)`
`quad quad quad xx (cosA cosB + sinA sinB)`
`quad = cos^2 A cos^2B - sin^2A sin^2B`

substitute `cos^2 B = (1-sin^2 B)`
`quad = cos^2A - sin^2B xx(sin^2 A+cos^2 A)`
`quad = cos^2A - sin^2B`

substitute `sin^2B = (1-cos^2 B)`
`quad = cos^2B(cos^2A+sin^2A)-sin^2A`
`quad = cos^2 B - sin^2 A`

`cos(A+B)cos(A-B)`
`quad quad = cos^2 A - sin^2 B`
`quad quad = cos^2 B - sin^2 A`

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

and multiply numerators and denominators of them
`quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)`

`tan(A+B) tan(A-B)`
`quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)`

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

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


and multiply numerator and denominator
`quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)`

`cot(A+B) cot(A-B)`
`quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)`

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

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

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

`cos(A+B+C)`
`quad = cosA cosB cos C - sinA sinB cos C`
`quad quad quad - sinA cosB sinC - cosA sinB sinC `

More results of Trigonometric values for compound Angles `sin(A+B)sin(A-B)`
`quad quad = sin^2 A - sin^2 B`
`quad quad = cos^2 B - cos^2 A`

`cos(A+B)cos(A-B)`
`quad quad = cos^2 A - sin^2 B`
`quad quad = cos^2 B - sin^2 A`

`tan(A+B) tan(A-B)`
`quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)`

`cot(A+B) cot(A-B)`
`quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)`

`sin(A + B + C) = sin((A+B)+C)`

`cos(A + B + C) = cos((A+B)+C)`

                            
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