Question

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\( (\text{cosec θ}-\text{cot θ})²=\frac{(1-\text{cos θ})}{(1 +\text{cos θ})}\)

Answer 1

\( (\text{cosec θ}-\text{cot θ})²=\frac{(1-\text{cos θ})}{(1 +\text{cos θ})}\)

L.H.S =\( (\text{cosec θ - cot θ})²\)
\(= \left(\frac{1}{\text{sin θ}} - \frac{\text{cos θ}}{\text{sin θ}}\right)^²\)

\(= \frac{(1 - \text{cos θ})²}{(\text{sin θ})²}\)

\(= \frac{(1 - \text{cos θ})²}{\text{sin² θ}}\)

\(=\frac{ (1 - \text{cos θ})²}{(1 - \text{cos²θ})}  \)    (By Identity sin A + cos A = 1 Hence, 1 - cos A= sin A)

\(= \frac{(1 - \text{cos θ})²}{ (1 - \text{cos θ})(1 + \text{cos θ})}    \)       [Using a² - b² = (a + b) (a - b)]

\(=\frac{ (1 -\text{ cos θ})}{(1 + \text{cos θ})}\)
= RHS