SinAsin2A+sin3Asin6A/sinAcos2A+sin3Acos6A
To simplify the given expression, let's break it down step by step.
The given expression is:
sin(A)sin(2A) + sin(3A)sin(6A)
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sin(A)cos(2A) + sin(3A)cos(6A)
Step 1: Expand the trigonometric identities using the product-to-sum formulas.
sin(A)sin(2A) can be written as (1/2)[cos(2A-A)-cos(2A+A)]
sin(3A)sin(6A) can be written as (1/2)[cos(3A-6A)-cos(3A+6A)]
sin(A)cos(2A) can be written as (1/2)[sin(2A+A)+sin(2A-A)]
sin(3A)cos(6A) can be written as (1/2)[sin(3A+6A)+sin(3A-6A)]
Step 2: Simplify each term.
(1/2)[cos(2A-A)-cos(2A+A)] simplifies to (1/2)(cos(A)-cos(3A))
(1/2)[cos(3A-6A)-cos(3A+6A)] simplifies to (1/2)(cos(-3A)-cos(9A))
(1/2)[sin(2A+A)+sin(2A-A)] simplifies to (1/2)(sin(3A)+sin(A))
(1/2)[sin(3A+6A)+sin(3A-6A)] simplifies to (1/2)(sin(9A)+sin(-3A))
Step 3: Rearrange the terms and combine like terms.
[(1/2)(cos(A)-cos(3A)) + (1/2)(cos(-3A)-cos(9A))] / [(1/2)(sin(3A)+sin(A)) + (1/2)(sin(9A)+sin(-3A))]
Combine the cosine terms:
= [(1/2)(-cos(3A)+cos(A)) - (1/2)(cos(9A)-cos(-3A))] / [(1/2)(sin(3A)+sin(A)) + (1/2)(sin(9A)+sin(-3A))]
Combine the sine terms:
= [(1/2)(-cos(3A)+cos(A)) - (1/2)(cos(9A)-cos(-3A))] / [(1/2)(2sin(A)cos(3A)) + (1/2)(2sin(-3A)cos(9A))]
Combine the common factors:
= [-cos(3A)+cos(A) - cos(9A)+cos(-3A)] / [sin(A)cos(3A) - sin(-3A)cos(9A)]
Step 4: Use the trigonometric identities.
cos(-x) = cos(x) --> cos(-3A) = cos(3A)
cos(180°-x) = -cos(x) --> cos(180°-9A) = -cos(9A)
= [-cos(3A)+cos(A) - cos(9A)-cos(3A)] / [sin(A)cos(3A) + sin(3A)cos(9A)]
Combine the terms with similar signs:
= [cos(A)-2cos(3A)] / [sin(A)cos(3A) + sin(3A)cos(9A)]
We have now simplified the given expression to:
[cos(A)-2cos(3A)] / [sin(A)cos(3A) + sin(3A)cos(9A)]
using your product-to-sum formulas,
sinA sin2A = 1/2 (cosA-cos3A)
sin3A sin6A = 1/2 (cos3A-cos9A)
and so on ...
so your fraction now is
(cosA-cos3A)+(cos3A-cos9A)
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(cos3A-sinA)+(sin9A-sin3A)
= (cosA-cos9A)/(sin9A-sinA)
now go back the other way using sun-to-product formulas, and simplify