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Thread: trigonometry (3)

  1. #1
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    trigonometry (3)

    If $\displaystyle 0<A<\frac{\pi}{2}$ , and $\displaystyle 0<B<\frac{\pi}{2}$ , prove that

    (1)$\displaystyle
    \frac{1}{\cos^2 A}+\frac{1}{\sin^2 A\sin^2 B\cos^2 B}\geq 9
    $

    (2) Prove that $\displaystyle sin^2 A + sin^2 B > sin A + sin B + sin Asin B -1 $
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) We have

    $\displaystyle \frac{1}{\sin^2A\sin^2B\cos^2B}=\frac{\sin^2B+\cos ^2B}{\sin^2A\sin^2B\cos^2B}=\frac{1}{\sin^2A\cos^2 B}+\frac{1}{\sin^2A\sin^2B}$

    Then we have to prove that

    $\displaystyle \frac{1}{\cos^2A}+\frac{1}{\sin^2A\cos^2B}+\frac{1 }{\sin^2A\sin^2B}\geq 9$

    Use the inequality $\displaystyle (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\r ight)\geq 9$

    for $\displaystyle a=\cos^2A, \ b=sin^2A\cos^2B, \ c=\sin^2A\sin^2B$

    using that

    $\displaystyle a+b+c=\cos^2A+\sin^2A\cos^2B+\sin^2A\sin^2B=$

    $\displaystyle =\cos^2A+\sin^2A(\sin^2B+\cos^2B)=\cos^2A+\sin^2A= 1$
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  3. #3
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    Quote Originally Posted by thereddevils View Post
    If $\displaystyle 0<A<\frac{\pi}{2}$ , and $\displaystyle 0<B<\frac{\pi}{2}$ , prove that

    (1)$\displaystyle
    \frac{1}{\cos^2 A}+\frac{1}{\sin^2 A \sin^2 B \cos^2 B} \geq 9
    $

    (2) Prove that $\displaystyle sin^2 A + sin^2 B > sin A + sin B + sin Asin B -1 $

    For (1) $\displaystyle \sec^2 A + \text{cosec}^2 A \, \text{cosec}^2 B \, \sec^2 B$

    $\displaystyle =(1+\tan^2 A)+(1+\cot^2 A)(1+\cot^2 B)(1+tan^2 B)$

    $\displaystyle =(1+\tan^2 A)+(1+\cot^2 A)(2+\tan^2 B+\cot^2 B)$

    Consider $\displaystyle (\tan B-\cot B)^2\geq 0 $

    $\displaystyle \tan^2 B+\cot^2 B \geq 2$

    Then $\displaystyle (\tan A-2\cot A)^2\geq 0$

    $\displaystyle \tan^2 A+4\cot^2 A\geq 4$

    continuing

    $\displaystyle =(1+\tan^2 A)+(1+\cot^2 A)(4)$

    $\displaystyle =5+\tan^2 A+4\cot^2 A$

    $\displaystyle =9$

    this inequality is at least 9 . Hence , proved .
    Last edited by mr fantastic; Oct 8th 2009 at 04:04 AM. Reason: Fixed first line of latex: \cosec is not a valid command. Use \text{cosec}.
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