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Question

# $\mathrm{sin}5x=5{\mathrm{cos}}^{4}x\mathrm{sin}x-10{\mathrm{cos}}^{2}x{\mathrm{sin}}^{3}x+{\mathrm{sin}}^{5}x$

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Solution

## $\mathrm{LHS}=\mathrm{sin}5\mathrm{x}\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left(3x+2x\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}3x×\mathrm{cos}2x+\mathrm{cos}3x×\mathrm{sin}2x\phantom{\rule{0ex}{0ex}}=\left(3\mathrm{sin}x-4{\mathrm{sin}}^{3}x\right)\left(2{\mathrm{cos}}^{2}x-1\right)+\left(4{\mathrm{cos}}^{3}x-3\mathrm{cos}x\right)×2\mathrm{sin}x\mathrm{cos}x\phantom{\rule{0ex}{0ex}}=-3\mathrm{sin}x+4{\mathrm{sin}}^{3}x+6\mathrm{sin}x{\mathrm{cos}}^{2}x-8{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x+8\mathrm{sin}x{\mathrm{cos}}^{4}x-6\mathrm{sin}x{\mathrm{cos}}^{2}x\phantom{\rule{0ex}{0ex}}=8\mathrm{sin}x{\mathrm{cos}}^{4}x-8{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-3\mathrm{sin}x+4{\mathrm{sin}}^{3}x\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-3\mathrm{sin}x+3\mathrm{sin}x{\mathrm{cos}}^{4}x+4{\mathrm{sin}}^{3}x+2{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-3\mathrm{sin}x\left(1-{\mathrm{cos}}^{4}x\right)+2{\mathrm{sin}}^{3}x\left(2+{\mathrm{cos}}^{2}x\right)$ $=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-3\mathrm{sin}x\left(1-{\mathrm{cos}}^{2}x\right)\left(1+{\mathrm{cos}}^{2}x\right)+2{\mathrm{sin}}^{3}x\left(2+{\mathrm{cos}}^{2}x\right)\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-3{\mathrm{sin}}^{3}x\left(1+{\mathrm{cos}}^{2}x\right)+2{\mathrm{sin}}^{3}x\left(2+{\mathrm{cos}}^{2}x\right)\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x\left[3\left(1+{\mathrm{cos}}^{2}x\right)-2\left(2+{\mathrm{cos}}^{2}x\right)\right]\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x\left[3+3{\mathrm{cos}}^{2}x-4-2{\mathrm{cos}}^{2}x\right]\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x\left[{\mathrm{cos}}^{2}x-1\right]\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x×\left(-{\mathrm{sin}}^{2}x\right)\phantom{\rule{0ex}{0ex}}=5\mathrm{sin}x{\mathrm{cos}}^{4}x-10{\mathrm{sin}}^{3}x{\mathrm{cos}}^{2}x+{\mathrm{sin}}^{5}x\phantom{\rule{0ex}{0ex}}=5{\mathrm{cos}}^{4}x\mathrm{sin}x-10{\mathrm{cos}}^{2}x{\mathrm{sin}}^{3}x+{\mathrm{sin}}^{5}x\phantom{\rule{0ex}{0ex}}=\mathrm{RHS}\phantom{\rule{0ex}{0ex}}\mathrm{Hence}\mathrm{proved}.$

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