If x - 1-x - 2 cos- then x3 - 1-x3 -

We have x + 1/x = 2cosθ, Now x^3 + 1/x^3 = (x + 1/x)^3 3x 1/x(x + 1/x) = (2 cosθ)^3 3(2 cosθ) = 8 cos^3 θ 6 cosθ =2(4 cos^3 θ 3cosθ) = 2cos3θ. Tri ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

If x + 1/x = ...

Question

If x + $\frac{1}{x}$ = $2 c o s θ$ , then $x^{3}$ + $\frac{1}{x^{3}}$ =

$c o s 3 θ$

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$2 c o s 3 θ$

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$\frac{1}{2}$ $c o s 3 θ$

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$\frac{1}{3}$ $c o s 3 θ$

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Solution

The correct option is B
$2 c o s 3 θ$

We have x + $\frac{1}{x}$ = $2 c o s θ$ ,

Now $x^{3}$ + $\frac{1}{x^{3}}$ = ${(x + \frac{1}{x})}^{3}$ - $3 x \frac{1}{x} (x + \frac{1}{x})$

= $(2 c o s θ)^{3} - 3 (2 c o s θ) = 8 c o s^{3} θ - 6 c o s θ$

= $2 (4 c o s^{3} θ - 3 c o s θ) = 2 c o s 3 θ$ .

Trick: Put x = 1 $\Rightarrow$ $θ = 0^{\circ}$ .

Then $x^{3}$ + $\frac{1}{x^{3}}$ = 2 = $2 c o s 3 θ$ .

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Similar questions

Q. If

x + \frac{1}{x} = 2 cos θ,

then

x^{3} + \frac{1}{x^{3}} =

If x + $\frac{1}{x}$ = $2 c o s θ$ , then $x^{3}$ + $\frac{1}{x^{3}}$ =

[MP PET 2004]

Q. If

x + \frac{1}{x} = 2 cos θ,

then find the value of

x^{3} + \frac{1}{x^{3}} .

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Q. If

2 cos θ = x + \frac{1}{x}

, then prove that

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If x + 1x = 2cosθ, then x3 + 1x3 =

The correct option is B 2cos3θ We have x + 1x = 2cosθ, Now x3 + 1x3 = (x+1x)3 - 3x1x(x+1x) = (2cosθ)3−3(2cosθ)=8cos3θ−6cosθ =2(4cos3θ−3cosθ)=2cos3θ. Trick: Put x = 1 ⇒ θ=0∘. Then x3 + 1x3 = 2 = 2cos3θ.

If x + $\frac{1}{x}$ = $2 c o s θ$ , then $x^{3}$ + $\frac{1}{x^{3}}$ =