Question

I: $\cos 6\sin 24\cos 72=\frac{1}{8}$
II: $\cos 12\cos 24\cos 36\cos 48\cos 72\cos 84=\frac{1}{64}$
Note: $x$ in $\cos x$ and $\sin x$ is represented in degrees.

• A. only I is true
• B. only II is true
• C. Both I and II are true
• D. Neither I nor II are true

Answer 1

The correct option is C Both I and II are true

$b=\cos 6\sin 24\cos 72=\cos 6\sin (90-66)\cos 72$
$\Rightarrow b=\frac{\left(2\cos 6\cos 66\right)\cos 72}{2}$
$\Rightarrow b=\frac{(\cos (6+66)+\cos (6-66))\cos 72}{2}$
$\Rightarrow b=\frac{(\cos 72+\cos 60)\cos 72}{2}=\frac{2{\cos }^{2}72+\cos 72}{4}$
$\Rightarrow b=\frac{1+\cos 144+\cos 72}{4}=\frac{1+\cos (180-36)+\cos 72}{4}$
$\Rightarrow b=\frac{1+\cos 72-\cos 36}{4}=\frac{1-2\sin \left(\frac{72+36}{2}\right)\sin \left(\frac{72-36}{2}\right)}{4}$
$\Rightarrow b=\frac{1-2\sin 54\sin 18}{4}=\frac{1-2\cos 36\sin 18}{4}$
$\Rightarrow b=\frac{1-\left(\frac{2\sin 18\cos 18\cos 36}{\cos 18}\right)}{4}$
$\Rightarrow b=\frac{1-\frac{2\sin 36\cos 36}{2\cos 18}}{4}=\frac{1-\frac{\sin 72}{2\cos 18}}{4}$
$\Rightarrow b=\frac{1-\frac{\sin 72}{2\cos (90-72)}}{4}=\frac{1-\frac{\sin 72}{2\sin 72}}{4}$
$\Rightarrow b=\frac{1}{8}$
$a=\cos {12}^{}\cos {24}^{}\cos {36}^{}\cos {48}^{}\cos {72}^{}\cos {84}^{}$
$\Rightarrow a=\frac{\left(2\sin {12}^{}\cos {12}^{}\right)\cos {24}^{}\cos {36}^{}\cos {48}^{}\cos {72}^{}\cos {84}^{}}{2\sin {12}^{}}$
$\Rightarrow a=\frac{2\sin {24}^{}\cos {24}^{}\cos {36}^{}\cos {48}^{}\cos {72}^{}\cos {84}^{}}{4\sin {12}^{}}$
$\Rightarrow a=\frac{-2\sin {48}^{}\cos {48}^{}\cos {36}^{}\cos {72}^{}\cos ({180}^{}-{84}^{})}{8\sin {12}^{}}$
$\Rightarrow a=\frac{-2\sin {96}^{}\cos {96}^{}\cos {36}^{}\cos {72}^{}}{16\sin {12}^{}}$
$\Rightarrow a=\frac{-\sin {192}^{}\cos {36}^{}\cos {72}^{}}{16\sin {12}^{}}=\frac{-\sin ({180}^{}+{12}^{})\cos {36}^{}\cos {72}^{}}{16\sin {12}^{}}$
$\Rightarrow a=\frac{\sin {12}^{}\left(2\sin {36}^{}\cos {36}^{}\right)\cos {72}^{}}{32\sin {12}^{}\sin {36}^{}}=\frac{2\sin {72}^{}\cos {72}^{}}{64\sin {36}^{}}$
$\Rightarrow a=\frac{\sin {144}^{}}{64\sin {36}^{}}=\frac{\sin ({180}^{}-{144}^{})}{64\sin {36}^{}}=\frac{1}{64}$
Therefore, both the statements are correct.
Hence, option 'C' is correct.