Question

The value of $\frac{\cos {70}^o}{\sin {20}^o}+\frac{\cos {55}^o\text{cosec}{35}^o}{\tan 5^o\tan {25}^o\tan {45}^o\tan {65}^o\tan {85}^o}$ is :

• A. $1$
• B. $2$
• C. $0$
• D. $3$

Answer 1

Answer: (B)

$2$

$\frac{\cos {70}^o}{\sin {20}^o}+\frac{\cos {55}^o\text{cosec}{35}^o}{\tan 5^o\tan {25}^o\tan {45}^o\tan {65}^o\tan {85}^o}$

$=\frac{\cos ({90}^o-{20}^o)}{\sin {20}^o}+\frac{\cos {55}^o\text{cosec}({90}^o-{55}^o)}{\tan 5^o\tan {25}^o\tan {45}^o\tan ({90}^o-{25}^o)\tan ({90}^o-5^o)}$

$=\frac{\sin {20}^o}{\sin {20}^o}+\frac{\cos {55}^o\sec {55}^o}{\tan 5^o\tan {25}^o\tan {45}^o\cot {25}^o\cot 5^o}$

$=1+\frac{1}{\left(\tan 5^o\cot 5^o\right)\left(\tan {25}^o\cot {25}^o\right)\tan {45}^o}[\because \cos {55}^o\sec {55}^o=1]$
$=1+\frac{1}{\left(1\right)\left(1\right)1}=1+1=2$