Question

If cosec θ = ${\displaystyle \frac{29}{20}}$, find the value of: cosec θ - ${\displaystyle \frac{1}{\text{cot}\theta }}$

Answer 1

Consider ΔABC, where ∠A = 90°

⇒ cosec θ = ${\displaystyle \frac{\text{Hypotenuse}}{\text{Perpendicular}}=\frac{\text{BC}}{\text{AB}}=\frac{29}{20}}$

By Pythagoras theorem,
BC² = AB² + AC²
⇒ AC² = BC² - AB²
= 29² - 20²
= 841 - 400
= 441
⇒ AC = 21
Now,

sec θ = ${\displaystyle \frac{\text{Hypotenuse}}{\text{Base}}=\frac{\text{BC}}{\text{AC}}=\frac{29}{21}}$

tan θ = ${\displaystyle \frac{\text{Perpendicular}}{\text{Base}}=\frac{\text{AB}}{\text{AC}}=\frac{20}{21}}$

⇒ cot θ  = ${\displaystyle \frac{1}{\text{tan θ}}=\frac{21}{20}}$

${\displaystyle \text{cosec} \theta -\frac{1}{\text{cot θ}}}$

= ${\displaystyle \frac{29}{20}-\frac{1}{\frac{21}{20}}}$

= ${\displaystyle \frac{29}{20}-\frac{20}{21}}$

= ${\displaystyle \frac{609-400}{420}}$

= ${\displaystyle \frac{209}{420}}$.