Question

1/1+cos theta + 1/1-cos theta =2 cosec sq theta

Answer 1

$\bold{\text{To Prove:}}$

$\dfrac{1}{1 + \cos \theta } + \dfrac{1}{1 - \cos \theta } = 2 \cosec {}^{2} \theta$

$\text{Solution:}$

L.H.S

$\frac{1}{1 + \cos \theta } + \frac{1}{1 - \cos \theta } \\ \\ = \frac{1 - \cos \theta + 1 + \cos \theta }{(1 + \cos \theta )(1 - \cos \theta) } \\ \\ = \frac{1 - \cancel{\cos \theta} + 1 + \cancel{\cos \theta } }{1 {}^{2} - \cos {}^{2} \theta } \\ \\ = \frac{2}{1 - { \cos {}^{2} \theta}^{} } \\ \\ = \frac{2}{ \sin {}^{2} \theta } \\ \\ = 2 \cosec {}^{2} \theta \\ \\ \therefore \: L.H.S = R.H.S \: \: [Proved]$

$\bold{\text{Using Formula}}$

(a + b)(a - b) = a² - b²

$\boxed{ \bold{ \text{Some important trigonometry Rules: }}}$

sinø . cosecø = 1

cosø . secø = 1

tanø . cotø = 1

sin²ø + cos²ø = 1

cosec²ø - cot² = 1

sec²ø - tan²ø = 1