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Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

If sin A+si...

Question

If $s i n A + s i n^{2} A + s i n^{3} A = 1,$ then find the value of $c o s^{6} A - 4 c o s^{4} A + 8 c o s^{2} A$ .

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Solution

sinA + sin^2A + sin^3A = 1
sinA + sin^3A = 1 - sin^2A = cos^2A
sinA(1+sin^2A) = cos^2A
sinA(2 -cos^2A) = cos^2A
Squaring both sides,
sin^2A(4-4cos^2A +cos^4A) = cos^4A
(1-cos^2A)(4-4cos^2A +cos^4A) = cos^4A
4-4cos^2A +cos^4A-4cos^2A+4cos^A-cos^6A = cos^4A
4 -cos^6A +4cos^4A -8cos^2A = 0
cos^6A - 4cos^4A + 8cos^2A = 4

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Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

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