# A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere.

$$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$

Could someone please provide a valid proof?

I am not allowed to work on both sides of the equation.

Work so far:

RS:

\begin{align} & \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} + 2 \\[10pt] & = \frac{\sin^4\theta}{(\cos^2\theta)(\sin^2\theta)} + \frac{\cos^4\theta}{(\sin^2\theta) (\cos^2\theta)} + \frac{(\sin^4\theta)(\cos^2\theta)}{(\sin^2\theta)(\cos^2\theta)} + \frac{(\sin^2\theta)(\cos^4\theta)}{(\sin^2\theta)(\cos^2\theta)} \\[10pt] & = \frac{\sin^4\theta + \cos^4\theta + (\sin^4\theta)(\cos^2\theta) + (\sin^2\theta)(\cos^4\theta)}{(\cos^2\theta)(\sin^2\theta)} \end{align}

I am completely lost after this.

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Try to put everything on the same denominator. –  Raskolnikov May 7 '12 at 15:44
@Raskolnikov I tried that but I end up getting an extremely messy and large equation. After that, I have absolutely no idea how to proceed. –  user26649 May 7 '12 at 15:45
What happens if you multiply the whole thing by $\sin^2\theta\cos^2\theta$? –  anon May 7 '12 at 15:47
Ok, then please post that mess here.The problem is not messy at all.That way you can get feedback on where you went wrong. –  Amitabh Udayiman May 7 '12 at 15:51
@AmitabhUdayiman Posted. –  user26649 May 7 '12 at 16:04

Hint: $$(\tan\theta+\cot \theta)^2=\left(\frac{\sin\theta}{\cos \theta} +\frac{\cos \theta}{\sin \theta}\right)^2$$ $$= \left(\frac{\cos^2 \theta+\sin^2\theta}{\cos \theta \sin \theta}\right)^2.$$ Now try using the fact that $\cos^2\theta+\sin^2\theta=1$, twice.

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$\frac{1}{(\cos^2\theta)(sin^2\theta)}$ $=\frac{1}{cos^2\theta} * \frac{1}{sin^2\theta}$ @EricNaslund Why I arrive at a wrong equation? –  user26649 May 7 '12 at 16:16
$$\dfrac{1}{\cos^2(\theta) \sin^2(\theta)} = \dfrac{\cos^2 (\theta) + \sin^2 (\theta)}{\cos^2(\theta) \sin^2(\theta)} = \dfrac{\cos^2 (\theta)}{\cos^2(\theta) \sin^2(\theta)} + \dfrac{\sin^2 (\theta)}{\cos^2(\theta) \sin^2(\theta)}$$ –  The Chaz 2.0 May 7 '12 at 16:22
@TheChaz Thank You! Just wondering but since I arrived at a multiplicative equation, does that mean $\frac{1}{\cos^2\theta} * \frac{1}{\sin^2\theta} = \frac{1}{\cos^2\theta} + \frac{1}{\sin^2\theta}$ –  user26649 May 7 '12 at 16:25
You tell me! (If you add one more "$=$" to the end of my string of equalities, you'll have your result) :D –  The Chaz 2.0 May 7 '12 at 16:31
Thank Eric! It took me a while to see how to use that identity "twice" :) –  The Chaz 2.0 May 7 '12 at 16:37
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Look at the largest triangle.

(There's a reason my avatar --the logo of my software company-- is a stylized version of this figure.)

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Why have I never seen a diagram of a trig identity before?? (Besides the ones that should be memorized...) +1 –  The Chaz 2.0 May 7 '12 at 16:02
Very nice diagram, +1. –  Eric Naslund May 7 '12 at 17:44
I like a good geometric proof without words (for example). (+1) –  robjohn May 7 '12 at 20:40

Hint:

$$\tan(\theta) + \cot(\theta)= \frac{\sin(\theta)}{\cos(\theta)}+\frac{\cos(\theta)}{\sin(\theta)}=\frac{\sin^2(\theta)}{\sin(\theta)\cos(\theta)}+\frac{\cos^2(\theta)}{\sin(\theta)\cos(\theta)}$$

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$$\sin^2 \theta=\frac{\tan^2 \theta}{1+\tan^2 \theta}$$
$$\cos^2 \theta=\frac{1}{1+\tan^2 \theta}$$
$$\tan \theta = \frac{1}{\cot \theta}$$