<s>[INST] Prove that $6(\sin^{10}A+\cos^{10}A)-15(\sin^8A+\cos^8A)+10(\sin^6A+\cos^6A)-1=0​$ Prove that
​$$6(\sin^{10}A+\cos^{10}A)-15(\sin^8A+\cos^8A)+10(\sin^6A+\cos^6A)-1=0​$$
Expression can be verified for different values of $A$ such as $\frac\pi4,\frac\pi2$ etc. But to prove it for general value of A?
 [/INST] Hint: Try to use $\cos^2 A = 1-\sin^2 A $

 Writing $s$ for $\sin^2 A$ we have $6(s^5 + (1-s)^5) -15(s^4+ (1-s)^4) + 10(s^3 + (1-s)^3) - 1 = 6s^5 + 6 - 30s + 60s^2 - 60s^3 + 30s^4 - 6s^5 - 15s^4 - 15 + 60s - 90s^2 + 60s^3 - 15s^4 + 10s^3 + 10 - 30s + 30s^2 - 10s^3 - 1 = 0$

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