Let - and - be non zero real numbers such that 2cos-cos-cos- cos-1- Then which of the following is-are true-

We have, 2 (cos⁡β cos⁡α) +cos⁡⁡α cos⁡β=1 Or 4( cos⁡β cos⁡⁡α)+2 cos⁡⁡α cos⁡β=2 ⇒ 1 cos⁡⁡α+cos⁡β cos⁡α cos⁡β = 3+3 cos⁡α 3 cos⁡β 3 cos⁡ cos⁡β ⇒ (1 cos⁡⁡α) ...

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Byju's Answer

Standard XIII

Mathematics

Trigonometric Equations

Let α and β b...

Question

Let $α$ and $β$ be non zero real numbers such that $2 (c o s β - c o s α) + c o s α c o s β = 1$ . Then which of the following is/are true?

\sqrt{3} t a n (\frac{α}{2}) - t a n (\frac{β}{2}) = 0

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t a n (\frac{α}{2}) - \sqrt{3} t a n (\frac{β}{2}) = 0

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t a n (\frac{α}{2}) + \sqrt{3} t a n (\frac{β}{2}) = 0

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\sqrt{3} t a n (\frac{α}{2}) + t a n (\frac{β}{2}) = 0

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Solution

The correct option is C $t a n (\frac{α}{2}) + \sqrt{3} t a n (\frac{β}{2}) = 0$
We have, $2 (c o s β - c o s α) + c o s α c o s β = 1$
Or $4 (c o s β - c o s α) + 2 c o s α c o s β = 2$
$\Rightarrow 1 - c o s α + c o s β - c o s α c o s β$
$= 3 + 3 c o s α - 3 c o s β - 3 c o s α α c o s β$
$\Rightarrow (1 - c o s α) (1 + c o s β) = 3 (1 + c o s α) (1 - c o s β)$
$\Rightarrow \frac{(1 - c o s α)}{(1 + c o s α)} = \frac{3 (1 - c o s β)}{(1 + c o s β)}$
$\Rightarrow t a n^{2} \frac{α}{2} = 3 t a n^{2} \frac{β}{2}$
$∴ t a n \frac{α}{2} \pm \sqrt{3} t a n \frac{β}{2} = 0$

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Let α and β be non zero real numbers such that 2(cosβ−cosα)+cosαcosβ=1. Then which of the following is/are true?

Let $α$ and $β$ be non zero real numbers such that $2 (c o s β - c o s α) + c o s α c o s β = 1$ . Then which of the following is/are true?