Question

1/1+cos+1/1-cos=2cosec^2​

Answer 1

To prove :

$\star \: \frac{1}{1 + \cos( \alpha ) } + \frac{1}{1 - \cos( \alpha ) } = 2 { \csc }^{2} \alpha$

LHS :

$\implies \: \frac{1}{1 + \cos( \alpha ) } + \frac{1}{1 - \cos( \alpha ) } \\ \\ \implies \: \frac{1 - \cos( \alpha ) + 1 + \cos( \alpha ) }{ {1}^{2} - { \cos}^{2} \alpha } \\ \\ \implies \: \frac{2}{ { \sin}^{2} \alpha } \\ \\ \implies \: 2 { \csc }^{2} \alpha$

LHS = RHS

Proved .

Formula :

• sin² @ + cos² @= 1

• sin @ = 1/ csc@.

Answer 2

ɢɪᴠᴇɴ:-

• $\sf\dfrac{1}{1+cosA} + \dfrac{1}{1-cosA}$
• $\sf2{cosec}^{2}A$

ᴛᴏ ᴘʀᴏᴠᴇ:-

$\sf\dfrac{1}{1+cosA} + \dfrac{1}{1-cosA} = 2{cosec}^{2}A$

ᴘʀᴏᴏꜰ:-

LHS,

$:\implies \: \: \: \: \: \: \:\sf \dfrac{1}{1+cosA} + \dfrac{1}{1-cosA}1+cosA$

$:\implies \: \: \: \: \: \: \:\sf \dfrac{1- cosA + 1 + cosA}{(1+cosA)(1-cosA)}$

$:\implies \: \: \: \: \: \: \:\sf \dfrac{2}{1 - {cos}^{2}A}$

$:\implies \: \: \: \: \: \: \:\sf \dfrac{2}{{sin}^{2}A}$

$:\implies \: \: \: \: \: \: \:\sf 2{cosec}^{2}A$

ʜᴇɴᴄᴇ, ᴘʀᴏᴠᴇᴅ

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Identities used:-

• $\sf a^{2} - b^{2} = (a+b)(a-b)$
• $\sf{sin}^{2}\theta + {cos}^{2}\theta = 1$
• $\sf cosec\theta = \dfrac{1}{sin\theta}$