SATHEE: Trigonometrical Equations 1 Question 19

Trigonometrical Equations 1 Question 19

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

(2017 Adv.)

(a) $\sqrt{3} \tan \frac{α}{2} - \tan \frac{β}{2} = 0$

(b) $\tan \frac{α}{2} - \sqrt{3} \tan \frac{β}{2} = 0$

(d) $\sqrt{3} \tan \frac{α}{2} + \tan \frac{β}{2} = 0$

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Answer:

Correct Answer: 19. (b, c)

Solution:

We have, $2 (\cos β - \cos α) + \cos α \cos β = 1$

$\begin{array}{lc} or 4 (\cos β - \cos α) + 2 \cos α \cos β = 2 \\ \begin{array}{lc} \Rightarrow & 1 - \cos α + \cos β - \cos α \cos β \\ = 3 + 3 \cos α - 3 \cos β - 3 \cos α \cos β \\ \Rightarrow & (1 - \cos α) (1 + \cos β) = 3 (1 + \cos α) (1 - \cos β) \\ \Rightarrow & \frac{(1 - \cos α)}{(1 + \cos α)} = \frac{3 (1 - \cos β)}{1 + \cos β} \\ \Rightarrow & \tan^{2} \frac{α}{2} = 3 \tan^{2} \frac{β}{2} \\ ∴ & \tan \frac{α}{2} \pm \sqrt{3} \tan \frac{β}{2} = 0 \end{array} \end{array}$

Trigonometrical Equations 1 Question 19

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

Answer:

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Trigonometrical Equations 1 Question 19

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

Answer:

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