Question

If Sin A+sin^3A=cos^2A,prove that cos^6-4cos^4+8cos^2A=4?................

Answer 1

The proof is given below..........

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Answer 2

Answer:

$If \:Sin A+sin^{3}A=cos^{2}A\:then\:\\cos^{6}-4cos^{4}A+8cos^{2}A=4$

Step-by-step explanation:

$Given,\\sinA+sin^{3}A=cos^{2}A$

$\implies sinA(1+sin^{2}A)=cos^{2}A$

/* On Squaring both sides of the equation,we get

$\implies sin^{2}A\left(1+sin^{2}A\right)^{2}=cos^{4}A$

$\implies (1-cos^{2}A)[1+(1-cos^{2}A)]^{2}=cos^{4}A$

$\implies (1-cos^{2}A)(2-cos^{2}A)^{2}=cos^{4}A$

$\implies (1-cos^{2}A)(4-4cos^{2}A+cos^{4}A)=cos^{4}A$

$\implies 4-4cos^{2}A+cos^{4}A-4cos^{2}A+4cos^{4}A-cos^{6}A=cos^{4}A$

$\implies -cos^{6}A+4cos^{4}A-8cos^{2}A+4=0$

$\implies cos^{6}A-4cos^{4}A+8cos^{2}A=4$

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