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Answer on Calculus Question for Don

Question #5485
using MacLaurin Series: by mutiplying the series sinx and cosx and mutiplying the answer by 2, show that sin2x= 2sinxcosx?


please help me as soon as possible( thank you)
Expert's answer
Series representation of sine and cosine is given below:
<img style="width: 297px; height: 63px;" src="data:image/png;base64,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" alt="">
Hence multiplying these two expressions we obtain
<img style="width: 514px; height: 64px;" src="data:image/png;base64,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" 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<img style="width: 499px; height: 56px;" 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