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Can you please help me prove that this equation is true? $$\left(1- \frac{2\tan(x)}{\sin(2x)}\right)^2 = \left(1- \frac{2\tan(x)}{\tan(2x)}\right)^2$$

AtKin
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Hashem
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Aug 19 '22 at 16:53
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  • @user2661923 It's correct. Without squaring both sides, it is not. – Ted Shifrin Aug 19 '22 at 17:09
  • Sin Cos equation but there is no Cos? – peterwhy Aug 19 '22 at 17:09

1 Answers1

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I am assuming you are familiar with half and double angle formulae(especially manipulation in terms of $\tan(\theta/2)$, we shall be using the following formulae for the proof :-

$\sin2x = \frac{2\tan x}{1+\tan^2(x)} $
$\tan2x = \frac{2\tan x}{1-\tan^2(x)}$

Now to prove the following relation,

  1. Substitute $\sin2x$ and $\tan2x$ on the LHS and RHS of the equation

  2. We notice that the $2\tan x$ terms get cancelled leaving us with: $(1-(1+\tan^2(x))^2 = (1-(1-\tan^2(x))^2$

  3. Opening the parentheses, this simplifies to: $(-\tan^2(x))^2 = (\tan^2(x))^2$

  4. We may absorb the negative sign on the LHS, leaving us with: $\tan^4(x) = \tan^4(x)$

    Hence proving that the relation is true.

    Hope this helps!

InanimateBeing
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MrKaushik
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