Verify:   2tan(theta)/1+tan^2(theta) = sin2(theta)

Expert Answers

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You should use the following identity such that:

`1 + tan^2 theta = 1/(cos^2 theta)`

Substituting`1/(cos^2 theta)`  for `1 + tan^2 theta`  yields:

`(2 tan theta)/(1/(cos^2 theta)) = sin 2 theta`

You need to write tangent function using the fraction `sin theta/cos theta`  such that:

`2(sin theta/cos theta )*cos^2 theta = sin 2 theta`

Reducing by `cos theta`  yields:

`2 sin theta*cos theta = sin 2 theta`

Notice that the left side expresses the expansion of sine of double angle, hence the given identity `(2 tan theta)/(1 + tan^2 theta ) = sin 2 theta ` is checked.

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