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Using an isosceles triangle and basic trigonometric identities to prove the trigonometric double angle formulas

First of all, we construct an Isosceles Triangle like this:

With $$AB=BC=a$$ (say). And $$B=2x$$. Then $$AD \perp BC$$ is constructed.

Subsequently, since the angles $$BAC$$ and $$ACB$$ are equal, angle $$ACB = 90 - x$$, then angle $$CAD=x$$.

From triangle $$ADC$$:

$$\tan \ x\ =\ \frac{CD}{AD}$$

$$\implies \tan \ x\ =\ \frac{a-a\cos 2x}{a\sin 2x}$$

$$\implies \tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}$$

$$\implies \cos 2x=1-\sin 2x\tan x$$

Squaring both sides,

$$\implies \cos ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x$$

$$\implies 1-\sin ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x$$

$$\implies \sin ^22x=2\sin 2x\tan x-\sin ^22x\tan ^2x$$

$$\sin 2x$$ is not necessarily $$0$$, so we can divide both sides by $$\sin 2x$$.

$$\implies \sin 2x=2\tan x-\sin 2x\tan ^2x$$

$$\implies \sin 2x\left(1+\tan ^2x\right)=2\tan x$$

$$\implies \boxed{\sin 2x=\frac{2\tan x}{1+\tan ^2x}}$$

Furthermore,

$$\sin 2x=\frac{2\tan x}{1+\tan ^2x}=\frac{2\tan x}{\sec ^2x}=2\tan x\cos ^2x=2\frac{\sin x}{\cos x}\cos ^2x=\boxed{2\sin x\cos x}$$

Now, from $$\tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}$$:

$$\sin 2x=\ \frac{1-\cos 2x}{\tan x}$$

$$\implies \frac{2\tan x}{1+\tan ^2x}=\ \frac{\left(1-\cos 2x\right)}{\tan x}$$

$$\implies \frac{2\tan ^2x}{1+\tan ^2x}=\ 1-\cos 2x$$

$$\implies \cos 2x=1-\frac{2\tan ^2x}{1+\tan ^2x}$$

$$\implies \boxed{\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x}}$$

From here,

$$\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x} = \frac{\cos ^2x\left(1-\tan ^2x\right)}{\cos ^2x\left(1+\tan ^2x\right)} = \frac{\cos ^2x-\sin ^2x}{\cos ^2x+\sin ^2x} = \frac{\cos ^2x-\sin ^2x}{1} = \boxed{\cos ^2x-\sin ^2x}$$

Combining the expansions,

$$\tan 2x=\frac{\sin 2x}{\cos 2x}=\frac{\left(\frac{2\tan x}{1+\tan ^2x}\right)}{\left(\frac{1-\tan ^2x}{1+\tan ^2x}\right)} = \boxed{\frac{2\tan x}{1-\tan ^2x}}$$

Note by Arkajyoti Banerjee
7 months, 1 week ago

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All is to confusing

- 7 months ago

Yeah, it is but an indirect way of deriving the identities using just basic trigonometry.

- 7 months ago

i can not understand anything. explanation anyone??????

- 7 months ago

What did you not understand, the construction of and in the triangle?

- 7 months ago