Show thet: 3( - )+6( + )^2+4( ^6x+ ^6x)=13

Question

Show thet: $3(\sin\text{x}-\cos\text{x})+6(\sin\text{x}+\cos)^2+4(\sin^6\text{x}+\cos^6\text{x})=13$

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Answer

$\text{LHS}=3(\sin\text{x}-\cos\text{x})+6(\sin\text{x}+\cos)^2+4(\sin^6\text{x}+\cos^6\text{x})$ $=3[\sin^4\text{x}-4\sin^3\text{x}\cos\text{x}+6\sin^2\text{x}\cos^2\text{x}-4\sin\text{x}\cos^3\text{x}+\cos^4\text{x}]$ $+6[\sin^2\text{x}+2\sin\text{x}\cos\text{x}+\cos^2\text{x}]+4(\sin^6\text{x}+\cos^6\text{x})$ $\big[\because(\text{a}-\text{b})^4=\text{a}^4-4\text{a}^3\text{b}+6\text{a}^2\text{b}^2-4\text{ab}^3+\text{b}^4$ by binomial expainsion$\big]$ $=3\big[\sin^4\text{x}+\cos^4\text{x}-4\sin\text{x}\cos\text{x}(\sin^2\text{x}+\cos^2\text{x})+6\sin^2\text{x}\cos^2\text{x}\big]$ $+6[1+2\sin\text{x}\cos\text{x}]+4\big[(\cos^2\text{x}+\sin^2\text{x})(\cos^4\text{x}-\cos^2\text{x}+\sin^4\text{x})\big]$ $\big[\because\text{a}^3+\text{b}^3=(\text{a+b})(\text{a}^2-\text{ab+b}^2)$ $=7[\sin^4\text{x}+\cos^4\text{x}]+18\sin^2\text{x}\cos^4\text{x}-4\sin^2\text{x}\cos^2\text{x}+6$ $=7[\sin^4\text{x}+\cos^4\text{x}2\sin^2\cos^2]+6$ $=7[\sin^2\text{x}+\cos^2\text{x}+2\sin^2\text{x}\cos^2\text{x}]+6$ $=7+6$ $=13=\text{RHS}$

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