David R.

asked • 06/01/20

Tan Half Angle Formula Question

tan(x/2) = (1-cosx)/sinx


tan2x = 2tanx/(1-tan^2x)

tanx = 2tan(x/2)/(1-tan^2x/2)


tanx * tan^2x/2 + 2tanx/2 - tanx = 0


Using quadratic formula

tanx/2 = [-1 +- radical(1+tan^2x)]/tanx

= (-1 += secx)/tanx

= (-cosx+-1)/sinx


I ended up getting two answers, but it seems like only one works. Can anyone explain to me why it is happening and when I should check LHS and RHS?

1 Expert Answer

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Sava D. answered • 06/02/20

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Using algebra to prove trigonometric identities is very difficult. After you use the formula, you need some more exercises to find the answer.


Workaround: use half angle formulas and trigonometric identities.

We will start with RHS (right hand side) and will transform it into LHS (left hand side).

(1 - cos(x))/sin(x)

= [(sin2(x/2) + cos2(x/2)) - (cos2(x/2) - sin2(x/2))] / [(2 sin(x/2) x cos(x/2)] =

...

= tan(x/2).


I suggest use your algebra skills to try to simplify the above expression to get tan(x/2).



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Sava D.

tutor
I still need to answer the question posed. The quadratic equation has two solutions. Since it is not a perfect square, only one of the solutions will be valid. You need to see which one, and you need to discard the other one by showing that it is not valid. This is the reason not to use quadratic equation to prove trigonometric identity. If you want to find solution of the quadratic, you can use the following substitution tan(x/2) = z; tan(x) = sin(x)/cos(x). Your equation is sin(x)/cos(x) = 2z/(1-z^2); sin(x)(1 - z^2) = 2 cos(x) z; ... z_1 = (1 - cos(x))/sin(x); z_2 = -(1 + cos(x))/sin(x). the second solution simplifies to z_2 = - cos(x/2)/sin(x/2).
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06/02/20

David R.

Thanks a lot for your comment. Could you please explain to me why only one of the solutions will be valid when it is not a perfect square?
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06/03/20

Sava D.

tutor
A quadratic usually has two distinct solutions. In the general case, the trigonometric quadratic equation will have two different expressions for tan(x/2). However, tan(x/2) is a function and it has only one value for any given value of x. For that reason, when we have two solutions, only one of them will match tan(x/2). The conclusion I reach is that when proving trigonometric identities, avoid using quadratic equations to prove them. You still will need to use trig identities and relationships to do so.
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06/03/20

David R.

Totally makes sense. Thanks a lot!
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06/03/20

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