Question

Cot 2x + tan x= cosec 2x

Answer 1

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Answer 2

$cot\ 2x + tan\ x = cosec\ 2x$

Solution:

Given that, we have to prove,

$cot\ 2x + tan\ x = cosec\ 2x$

Take the L.H.S

$cot\ 2x + tan\ x\\\\We\ know\ that\ \\\\\boxed{ cot\ x = \frac{cos\ x}{sin\ x} }$

$Therefore\\\\\frac{cos\ 2x}{sin\ 2x} + tan\ x$

$We\ know\ that \\\\\boxed{ tan\ x = \frac{sin\ x}{cos\ x} }$

$Therefore\\\\\frac{cos\ 2x}{sin\ 2x} + \frac{sin\ x}{cos\ x}$

$Simplify\\\\\frac{cos\ 2x\ cos\ x + sin\ x sin\ 2x }{ sin\ 2x cos\ x }$

$We\ know\ that\\\\cos(a - b) = cos\ a cos\ b + sin\ a sin\ b\\\\Therefore$

$\frac{ cos (2x - x ) }{ sin\ 2x cos\ x }\\\\\frac{ cos\ x}{ sin\ 2x cos\ x}\\\\\frac{1}{sin\ 2x}$

$We\ know\ that\\\\\boxed { cosec\ x = \frac{1}{sin\ x}}\\\\Therefore\\\\\frac{1}{sin\ 2x} = cosec\ 2x\\\\\Thus\\\\LHS = RHS\\\\Thus\ proved$

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