2 tan x

1+ tan^{2 }x

2 tan x

1+ tan^{2 }x

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Before solving this problem, it's useful to know how the three Pythagorean identities arise from the Pythagorean formula.

If you have a right triangle, select one of the non-right angles as *x*, and label the three sides as
*adj*, *opp*, and *hyp* to represent the lengths of the sides adjacent to
*x*, opposite from *x*, and the length of the hypotenuse. Then Pythagorean theorem states:

*opp*^{2} + *adj*^{2} = *hyp*^{2}

Divide both sides by *hyp*^{2} to obtain:

(*opp*/*hyp*)^{2} + (*adj*/*hyp*)^{2} = 1

By substituting sin *x* = *opp*/*hyp *and cos *x* =
*adj*/*hyp* , we get the most important trigonometric identity:

sin^{2} *x* + cos^{2} *x* = 1.

If you've learned circular trigonometry then I recommend the website: http://www.mathsisfun.com/geometry/unit-circle.html for an interactive demonstration, and the site: http://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html has a great interactive unit circle.

I don't even bother memorizing the other two Pythagorean identities. I just derive them when I need them by dividing both sides of that most important trigonometric identity by cos^{2}
*x* or sin^{2} *x*. Dividing by cos^{2} *x* gives us:

(sin *x* / cos *x*)^{2} + 1 = (1 / cos *x*)^{2}

Using the definitions for tangent and secant in terms of sine and cosine yields:

tan^{2} *x* + 1 = sec^{2 }*x*

Of course, if you like memorization better than derivation then memorize all three identities, but you still should understand how to derive them. Using this identity with your problem, we get:

(2tan *x*) / (1+tan^{2}* x*) = 2tan *x* / sec^{2}* x*

= 2(sin *x* / cos *x*) / (1/cos^{2} *x*)

= 2(sin *x* / cos *x*)(cos^{2} *x*)

= 2 sin *x* cos *x*

That's a fairly simple expression, isn't it? But using other trigonometric identities that you're expected to know, you should be able to think of a way to simplify it even further.

From Pythagorean identities:

1 + tan^2 x = sec^2 x

and

tan^2 x = sin^2 x/cos^2 x

So, now instead of

2 tan x / (1 + tan^2 x)

we have:

2(sin^2 x/cos^2 x) / sec^2 x

but 1/cos^2 x = sec^2 x

2(sin^2 x *sec^2 x) / sec^2 x

2(sin^2 x)

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## Comments

I think you may have changed "2 tan x" to "2 tan^2 x" in the middle of this solution, leading to an error.