Which of the following statement is/are correct?

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$\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15} = \frac{1}{2^{7}}$

$\cos (\log_{5} (\frac{\sin^{2} A + \cos^{2} A + \tan^{2} A - \sec^{2} A . \sin^{2} A}{(1 + \tan^{2} A) (1 - \sin^{2} A)})) = 1$ , whenever it’s defined

If $p = \cos 55^{o}$ , $q = \cos 65^{o}$ and $r = \cos 175^{o}$ , then the value of $\frac{1}{p} + \frac{1}{q} + \frac{r}{p q}$ is equal to 1

Suppose that for some angles x and y the equation cos2y – cos2x = 3a – 2 and $\cos^{2} x + \sin^{2} y = \frac{a^{2}}{2}$ hold simultaneously, then number of possible values of a is 1

answer is A, B, D.

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Detailed Solution

a) $= \frac{- 1}{2} (\frac{\sqrt{5} + 1}{4}) (\frac{\sqrt{5} - 1}{4}) \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15}$
$= \frac{- 1}{8} \times \frac{\sin \frac{16 π}{15}}{16 \sin \frac{π}{15}} = \frac{1}{128} = \frac{1}{2^{7}}$
b) $\cos [\log_{5} (\frac{1 + \tan^{2} A - \sec^{2} A . \sin^{2} A}{\sec^{2} A . \cos^{2} A})] = \cos [\log_{5} (\frac{\sec^{2} A (1 - \sin^{2} A)}{1})] = \cos 0 = 1$
$\frac{1}{p} + \frac{1}{q} + \frac{r}{p q} = \frac{p + q + r}{p q} = \frac{\cos 55^{o} + 2 \cos 120^{o} \cos 55^{o}}{p q} = 0$
d) $(2 \cos^{2} y - 1) - (2 \cos^{2} x - 1) = 3 a - 2$
$\cos^{2} y - \cos^{2} x = \frac{3 a - 2}{2} ...... (1)$
$\cos^{2} x + \sin^{2} y = \frac{a^{2}}{2} ...... (2)$
Equation (1) + (2)
$1 = \frac{a^{2} + 3 a - 2}{2} \Rightarrow a^{2} + 3 a - 4 = 0$
a = 1 & a = - 4 ( impossible)