SATHEE: Trigonometrical Ratios and Identities 1 Question 15

Trigonometrical Ratios and Identities 1 Question 15

15. $1 + \cos \frac{π}{8} 1 + \cos \frac{3 π}{8} 1 + \cos \frac{5 π}{8} 1 + \cos \frac{7 π}{8}$ is equal to

(a) $\frac{1}{2}$

(b) $\cos \frac{π}{8}$

(d) $\frac{1 + \sqrt{2}}{2 \sqrt{2}}$

$(1984, 3 M)$

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Answer:

Correct Answer: 15. (b)

Solution:

Given expression $=$

$\begin{aligned} 1 & + \cos \frac{π}{8} 1 + \cos \frac{3 π}{8} 1 + \cos \frac{5 π}{8} 1 + \cos \frac{7 π}{8} \\ = 1 + \cos \frac{π}{8} 1 + \cos \frac{3 π}{8} 1 - \cos \frac{3 π}{8} 1 - \cos \frac{π}{8} \end{aligned}$

$\begin{aligned} = 1 - \cos^{2} \frac{π}{8} 1 - \cos^{2} \frac{3 π}{8} \\ = \frac{1}{4} 2 - 1 - \cos \frac{π}{4} 2 - 1 - \cos 3 \frac{π}{4} \\ = \frac{1}{4} 1 - \cos \frac{π}{4} 1 - \cos 3 \frac{π}{4} \\ = \frac{1}{4} 1 - \frac{1}{\sqrt{2}} 1 + \frac{1}{\sqrt{2}} = \frac{1}{4} 1 - \frac{1}{2} = \frac{1}{8} \end{aligned}$

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Trigonometrical Ratios and Identities 1 Question 15

15. 1+cos⁡π81+cos⁡3π81+cos⁡5π81+cos⁡7π8 is equal to

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15. $1 + \cos \frac{π}{8} 1 + \cos \frac{3 π}{8} 1 + \cos \frac{5 π}{8} 1 + \cos \frac{7 π}{8}$ is equal to