Question

Assertion :Numerical value of expression
$x=4{\log }_{5}2+{\log }_5\sin \frac{\pi }{5}+{\log }_5\sin \frac{2\pi }{5}+{\log }_5\sin \frac{3\pi }{5}+{\log }_5\sin \frac{4\pi }{5}$ is $1$. Reason: $\sin {18}^o=\frac{1}{4}(\sqrt{5}-1)$ and $\cos {36}^o=\frac{1}{4}(\sqrt{5}+1)$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
• C. Assertion is correct but Reason is incorrect.
• D. Assertion is incorrect but Reason is correct.

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

We have, $x=4{\log }_{5}2+{\log }_5\sin \frac{\pi }{5}+{\log }_5\sin \frac{2\pi }{5}+{\log }_5\sin \frac{3\pi }{5}+{\log }_5\sin \frac{4\pi }{5}$
Using $\sin \frac{3\pi }{5}=\sin \frac{2\pi }{5};\sin \frac{4\pi }{5}=\sin \frac{\pi }{5}$ we get
$x=4{\log }_{5}2+{\log }_5\left({\sin }^2\frac{\pi }{5}{\sin }^2\frac{2\pi }{5}\right)[\because \log a+\log b+\log c+\log d=\log (abcd\left)\right]$
$x=4{\log }_{5}2+{\log }_5\left[(1-{\cos }^2{36}^o)(1-{\cos }^2{72}^o)\right]$
$x=4{\log }_{5}2+{\log }_5(1-{\cos }^2{36}^o)(1-{\sin }^2{18}^o)$

$x=4{\log }_{5}2+{\log }_5[1-\frac{1}{16}{(\sqrt{5}+1)}^2][1-\frac{1}{16}{(\sqrt{5}-1)}^2]$

$x=4{\log }_{5}2+{\log }_5\left[\frac{(10-2\sqrt{5})(10+2\sqrt{5})}{256}\right]$

$x=4{\log }_{5}2+{\log }_5\left(\frac{80}{256}\right)$
$x=4{\log }_{5}2+{\log }_5\left(\frac{5}{2^4}\right)$
$\Rightarrow x=4{\log }_{5}2+{\log }_{5}5-4{\log }_{5}2$
$\Rightarrow x=1$Hence, assertion is correct.
Also, we know that
$\sin {18}^o=\frac{1}{4}(\sqrt{5}-1)$ and $\cos {36}^o=\frac{1}{4}(\sqrt{5}+1)$
Hence, reason is the correct statement and is the correct explanation for assertion.