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Question

# Write the value of 2 (sin6 x + cos6 x) −3 (sin4 x + cos4 x) + 1.

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Solution

## $2\left({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2.1\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=-\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x+1\phantom{\rule{0ex}{0ex}}=-\left\{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x+2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right\}+1\phantom{\rule{0ex}{0ex}}=-{\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)}^{2}+1\phantom{\rule{0ex}{0ex}}=-1+1\phantom{\rule{0ex}{0ex}}=0$

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