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Prove the following: cotAcot4A+1cotAcot4A-1=cos3Acos5A - Mathematics and Statistics

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Question

Prove the following:

`(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1) = (cos3"A")/(cos5"A")`

Sum
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Solution

L.H.S. = `(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1)`

= `(cos"A"/(sin"A")*(cos4"A")/(sin4"A") + 1)/(cos"A"/(sin"A")*(cos4"A")/(sin4"A") - 1`

= `((cos"A"cos4"A"+sin"A"sin4"A")/(sin"A"sin4"A"))/((cos"A"cos4"A"-sin"A"sin4"A")/(sin"A"sin4"A"))`

= `(cos4"A" cos"A" + sin4"A" sin"A")/(cos4"A" cos"A" - sin4"A" sin"A")`

= `(cos(4"A" - "A"))/(cos(4"A" + "A"))`

= `(cos3"A")/(cos5"A")`

= R.H.S.

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Chapter 3: Trigonometry - 2 - Exercise 3.1 [Page 39]

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