Question

If tanα=1/7, find the value of $\frac{ cosec^{2} \alpha - sec^{2} \alpha }{ cosec^{2} \alpha + sec^{2} \alpha }$

Answer 1

$\dfrac{\csc^2\alpha-\sec^2\alpha}{\csc^2\alpha+\sec^2\alpha}$

Using reciprocal identities $\csc\alpha = \frac{1}{\sin\alpha}, ~\sec\alpha=\frac{1}{\cos\alpha}$ the given expression becomes
$\dfrac{\frac{1}{\sin^2\alpha}-\frac{1}{\cos^2\alpha}}{\frac{1}{\sin^2\alpha}+\frac{1}{\cos^2\alpha}}$

Multiply top and bottom by $\sin^2\alpha$ and get
$\dfrac{1-\tan^2\alpha}{1+\tan^2\alpha}$

Plugin the given value and simplify
$\dfrac{1-\left(\frac{1}{7}\right)^2}{1+\left(\frac{1}{7}\right)^2}~=~\dfrac{49-1}{49+1}~=~\dfrac{48}{50}~=~\dfrac{24}{25}$