Question

if tan theta equals to 1 upon root 5 what is the value of cosec squared theta minus sec squared theta upon cos squared theta + sec squared theta ​

Answer 1

Answer:

$\bf{\frac{cosec^2\theta-sec^2\theta}{cos^2\theta+sec^2\theta}=\frac{144}{61}}$

Step-by-step explanation:

Given:

$tan\theta=\frac{1}{\sqrt5}$

$\implies\:cot\theta=\sqrt5$

$sec^2\theta=1+tan^2\theta$

$sec^2\theta=1+\frac{1}{5}$

$sec^2\theta=\frac{6}{5}$

$\implies\:cos^2\theta=\frac{5}{6}$

Now,

$\frac{cosec^2\theta-sec^2\theta}{cos^2\theta+sec^2\theta}$

$=\frac{(1+cot^2\theta)-(1+tan^2\theta)}{cos^2\theta+sec^2\theta}$

$=\frac{cot^2\theta-tan^2\theta}{cos^2\theta+sec^2\theta}$

$=\frac{5-\frac{1}{5}}{\frac{5}{6}+\frac{6}{5}}$

$=\frac{\frac{25-1}{5}}{\frac{25+36}{30}}$

$=\frac{\frac{24}{5}}{\frac{61}{30}}$

$=\frac{24}{5}*\frac{30}{61}}$

$=\frac{24}{1}*\frac{6}{61}}$

$=\frac{144}{61}$

Answer 2

Given : $\tan \theta = \frac{1}{\sqrt{5}}$

To find : $\frac{\csc^2\theta-\sec^2\theta}{\cos^2\theta+\sec^2\theta}$

Step-by-step explanation:

$\tan \theta = \frac{1}{\sqrt{5}}\\\\\cot \theta = \sqrt{5}\\\\\sec^2\theta= 1+ \tan^2\theta\\\\\sec^2\theta=1+\frac{1}{5}\\\\\sec^2\theta=\frac{6}{5}\\\\\cos^2\theta=\frac{5}{6}\\\\\frac{\csc^2\theta-\sec^2\theta}{\cos^2\theta+\sec^2\theta}\\\\\frac{(1-\cot^2\theta)-(1+\tan^2\theta)}{\cos^2\theta+\sec^2\theta}\\\\\frac{\cot^2\theta-\tan^2\theta}{\cos^2\theta+\sec^2\theta}\\\\\frac{5-\frac{1}{5}}{\frac{5}{6}+\frac{6}{5}}\\\\\frac{\frac{24}{5}}{\frac{61}{30}}\\\\\frac{24}{5}\times \frac{30}{61}$

$\\\\=\frac{144}{61}$

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