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What is the minimum value of this expression? $$\sin^2\theta+\cos^2\theta+\csc^2\theta+\sec^2\theta+\tan^2\theta+\cot^2 \theta$$

I tried grouping $\sin^2x+\csc^2x$, $\cos^2x+\sec^2x$ and $\cot^2x+\tan^2x$. I got the answer as $6$ but the book says $7$. How?

Michael Rozenberg
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mampu
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    You can minimise your three expressions separately, but they don't attain their minimum value at the same place. So the sum of the minima is less that the minimum of the sum. – Mark Bennet Sep 29 '18 at 12:00
  • Related : https://math.stackexchange.com/questions/1065943/to-prove-sin-theta-csc-theta2-cos-theta-sec-theta2-ge-9 – lab bhattacharjee Sep 29 '18 at 12:29

2 Answers2

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Since $\tan^2\theta=\frac{1}{\cos^2\theta}-1$ and $\cot^2\theta=\frac{1}{\sin^2\theta}-1$, your expression it's $$2\left(\frac{1}{\cos^2\theta}+\frac{1}{\sin^2\theta}\right)-1=\frac{8}{\sin^22\theta}-1\geq7.$$ The equality occurs for $\theta=45^{\circ}.$

Michael Rozenberg
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  • What I have done is that I have evaluated $Cos^2x+Sec^2x$ using AM and GM and hence I got the answer as $ 2$ .I have done the same for the other two.Hence I got $2+2+2$ which is $6$ – mampu Sep 29 '18 at 11:52
  • Post please your wrong solution. I'll find a mistake. – Michael Rozenberg Sep 29 '18 at 11:57
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$$\sin^2\theta+\cos^2\theta+\csc^2\theta+\sec^2\theta+\tan^2\theta+\cot^2\theta$$

$$=1+1+1+2(\tan^2\theta+\cot^2\theta)$$

Now $\tan^2\theta+\cot^2\theta=(\tan\theta-\cot\theta)^2+2\ge2$

lab bhattacharjee
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