Question

4.
1
= cosec 0 + cot O.
cosec - coto
Solution:​

Answer 1

Given to solve :-

$\sf{\dfrac{1}{cosec \: \theta - cot \: \theta} = cosec \: \theta + cot \: \theta}$

Answer :-

We have,

L.H.S = $\sf{\dfrac{1}{cosec \: \theta - cot \: \theta}}$

R.H.S = $\sf{cosec \: \theta + cot \: \theta}$

Therefore,

$\\ \sf{\longrightarrow \: \dfrac{1}{cosec \: \theta - cot \: \theta}}$

By multiplying both the numerator and denominator by (cosec θ + cot θ ), we get,

$\\ \sf{\longrightarrow \: \dfrac{1}{cosec \: \theta - cot \: \theta} \times \dfrac{cosec \: \theta + cot \: \theta}{cosec \: \theta + cot \: \theta}}$

By multiplying, we get,

$\\ \sf{\longrightarrow \: \dfrac{1(cosec \: \theta + cot \: \theta)}{(cosec \: \theta - cot \: \theta)(cosec \: \theta + cot \: \theta)}}$

By using (a - b)(a + b) = (a² - b²), We get,

$\\ \sf{\longrightarrow \: \dfrac{cosec \: \theta + cot \: \theta}{{(cosec \: \theta - cot \: \theta)}^{2}}}$

By using the identity (cosec θ - cot θ )² = 1, We get,

$\\ \sf{\longrightarrow \: \dfrac{cosec \: \theta + cot \: \theta}{1}}$

$\\ \sf{\longrightarrow \: cosec \: \theta + cot \: \theta = R.H.S}$

Hence, Proved!

⠀⠀⠀⠀⠀ ⠀⠀__________________

THINGS TO REMEMBER :-

• (cosec θ + cot θ)² = 1
• (a - b)(a + b) = (a² - b²).