Question

(sec^(2)θ )/(sec^(2)θ -1)

show steps

Answer 1

To Find:

$\boxed{ \sf \frac{ {(sec)}^{2} θ }{ {(sec)}^{2} θ - 1}}$

Solution:

we know that,

$\sf \red{ {(sec)}^{2} θ - 1 = {(tan)}^{2} θ}$

therefore,

$\sf \frac{ {sec}^{2} θ }{ {sec}^{2} θ - 1} = \frac{ {sec}^{2}θ }{ {tan}^{2}θ } \\ \sf \frac{ {sec}^{2}θ }{ {tan}^{2}θ } = \frac{1}{cos²θ} \times \frac{1}{tan²θ} \\ \sf = \frac{1}{cos²θ} \times cot²θ \\ \sf = \frac{1}{ \cancel{cos²θ}} \times \frac{ \cancel{cos²θ}}{sin²θ} \\ \sf = \frac{1}{sin² \: θ} = cosec² \: θ \\ \\ \boxed{ \bf \purple{ \frac{ {sec}^{2} θ }{ {sec}^{2} θ - 1} = cosec² θ}}$

#helpingismypleasure