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1/tana - 1/tan2a =cosec2a prove it ​

User Bibscy
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1 Answer

5 votes

Answer:

See below

Explanation:

To prove that
\sf (1)/(\tan ( a )) -( 1)/(tan(2a)) =cosec(2a) , we can start by expanding the left-hand side of the equation:


\sf (1)/(\tan ( a )) -( 1)/(tan(2a)) = (\cos ( a ))/(\sin ( a )) - (\cos ( 2a ))/(\sin(2a))

Using
\boxed{\sf tan(\theta) =(sin\theta)/(cos\theta)}

We can then use the identity
\sf \sin(2a) = 2\sin(a)\cos(a) to simplify the equation:


\sf = (\cos ( a ))/(\sin ( a )) - (\cos ( 2a ))/( 2\sin(a)\cos(a))

Taking L.C.M and simplifying it.


\sf = ( \cos(a) \cdot 2\cos(a) - \cos(2a) )/(2\sin(a)\cos(a))


\sf =( 2 \cos^2(a) - (cos^2(a) - sin^2(a) ) )/( 2\sin(a)\cos(a))


\sf = ( cos^2(a) + sin^2 (a) )/(2\sin(a)\cos(a))

Using Pythagorean identity
\sf cos^2(a) + \sin^2(a) = 1


\sf =(1)/(2\sin(a)\cos(a))

Again using identity property
\sf \sin(2a) = 2\sin(a)\cos(a)


\sf =(1)/(sin(2a))

Using
\sf sin \theta =(1)/(cosec \theta )


\sf =cosec (2a)

Since it is Right hand side.

Hence proved.

User John Day
by
7.9k points

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