If 2 cosθ = a +
1
a
, prove that 2 cos 2θ = a
2 +
1
a
2
and
2 cos 3θ = a
3 +
1
a
3

Question

If 2 cosθ = a + 1 a , prove that 2 cos 2θ = a 2 + 1 a 2 and 2 cos 3θ = a 3 + 1 a 3

Vikas TU · Answer

Dear Student,

2 cos theta = a+1/a

squaring both sides we get,(2 cos theta)^2=( a+1/a)^2

=>4 cos^2 theta=a^2+1/(a^2)+2

=>4 cos^2 theta-2=a^2+1/(a^2)

=>2(2cos^2 theta-1)=a^2+1/(a^2)

=>2 cos2 theta=a^2+1/(a^2) - hence first part is proved.

cubing both sides we get, (2 cos theta)^3=(a+1/a)^3

=>8cos^3 theta=a^3+1/(a^3)+3(a+1/a)

=>8cos^3 theta=a^3+1/(a^3)+3(2 cos theta) [from ques.]

=>8cos^3 theta-6 cos theta= a^3+1/(a^3)

=>2(4cos^3 theta - 3cos theta)=a^1/3 +1/(a^3)

=>2 cos3 theta =a^1/3 +1/(a^3) - hence second part is proved.

Cheers!!

Regards,

Vikas (B. Tech. 4^th year
Thapar University)

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If 2 cosθ = a + 1 a , prove that 2 cos 2θ = a 2 + 1 a 2 and 2 cos 3θ = a 3 + 1 a 3

If 2 cosθ = a +
1
a
, prove that 2 cos 2θ = a
2 +
1
a
2
and
2 cos 3θ = a
3 +
1
a
3

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If 2 cosθ = a + 1 a , prove that 2 cos 2θ = a 2 + 1 a 2 and 2 cos 3θ = a 3 + 1 a 3

If 2 cosθ = a +1a, prove that 2 cos 2θ = a2 +1a2and2 cos 3θ = a3 +1a3

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If 2 cosθ = a +
1
a
, prove that 2 cos 2θ = a
2 +
1
a
2
and
2 cos 3θ = a
3 +
1
a
3