Answer:
6g) We proved the equality [tex]cos4A=8cos^{4}A-8cos^{2}A+1[/tex] is true and hence proved.
Step-by-step explanation:
Given equation is [tex]cos4A=8cos^{4}A-8cos^{2}A+1[/tex]
To prove the equality LHS=RHS
[tex]cos4A=8cos^{4}A-8cos^{2}A+1[/tex]
Let us take LHS
[tex]cos4A=cos2(2A)[/tex]
[tex]=2cos^{2}2A-1[/tex] (since using [tex]cos2A=2cos^{2}A-1[/tex] here A=2A)
[tex]=2[(2cos^{2}2A-1)^2]-1[/tex]
[tex]=2[2^2cos^{4}2A-2\times 2cos^{2}2A+1]-1[/tex] (using the formula [tex](a-b)^2=a^2-2ab+b^2[/tex] here [tex]a=2cos^{2}2A[/tex] and b=1)
[tex]=2[4cos^{4}2A-4cos^{2}2A+1]-1[/tex]
[tex]=8cos^{4}2A-8cos^{2}2A+2-1[/tex]
[tex]=8cos^{4}2A-8cos^{2}2A+1[/tex]
[tex]=8cos^{4}2A-8cos^{2}2A+1=RHS[/tex]
[tex]cos4A=8cos^{4}2A-8cos^{2}2A+1=RHS[/tex]
Therefore LHS=RHS.
We proved the equality [tex]cos4A=8cos^{4}A-8cos^{2}A+1[/tex] is true and hence proved.
