Prove $sec^2 A + csc^2 A = 4 csc^2 2A$

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Hi would someone kindly help show me how to prove



$$sec^2 A + csc^2 A = 4 csc^2 2A$$



I got...



beginalignsec^2 A + csc^2 A
&= 1/cos^2 A + 1/sin^2 A
= (sin^2 A + cos^2 A) / [(cos^2 A)(sin^2 A)]\
&= 1 / [(cos^2 A)(sin^2 A)]endalign



but I am not sure on what algebraic or trigonometric trick is required to show RHS.



Thanks!




trigonometry secant




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  • Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
    – user401938
    Jul 19 at 21:39










  • "trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
    – DJohnM
    Jul 19 at 21:43















up vote
0
down vote

favorite












Hi would someone kindly help show me how to prove



$$sec^2 A + csc^2 A = 4 csc^2 2A$$



I got...



beginalignsec^2 A + csc^2 A
&= 1/cos^2 A + 1/sin^2 A
= (sin^2 A + cos^2 A) / [(cos^2 A)(sin^2 A)]\
&= 1 / [(cos^2 A)(sin^2 A)]endalign



but I am not sure on what algebraic or trigonometric trick is required to show RHS.



Thanks!




trigonometry secant




share|cite|improve this question





















  • Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
    – user401938
    Jul 19 at 21:39










  • "trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
    – DJohnM
    Jul 19 at 21:43













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Hi would someone kindly help show me how to prove



$$sec^2 A + csc^2 A = 4 csc^2 2A$$



I got...



beginalignsec^2 A + csc^2 A
&= 1/cos^2 A + 1/sin^2 A
= (sin^2 A + cos^2 A) / [(cos^2 A)(sin^2 A)]\
&= 1 / [(cos^2 A)(sin^2 A)]endalign



but I am not sure on what algebraic or trigonometric trick is required to show RHS.



Thanks!




trigonometry secant




share|cite|improve this question













Hi would someone kindly help show me how to prove



$$sec^2 A + csc^2 A = 4 csc^2 2A$$



I got...



beginalignsec^2 A + csc^2 A
&= 1/cos^2 A + 1/sin^2 A
= (sin^2 A + cos^2 A) / [(cos^2 A)(sin^2 A)]\
&= 1 / [(cos^2 A)(sin^2 A)]endalign



but I am not sure on what algebraic or trigonometric trick is required to show RHS.



Thanks!





trigonometry secant





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 19 at 21:59









RayDansh

884214




884214









asked Jul 19 at 21:36









number8

7717




7717











  • Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
    – user401938
    Jul 19 at 21:39










  • "trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
    – DJohnM
    Jul 19 at 21:43

















  • Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
    – user401938
    Jul 19 at 21:39










  • "trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
    – DJohnM
    Jul 19 at 21:43
















Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
– user401938
Jul 19 at 21:39




Use Mathjax please: math.meta.stackexchange.com/questions/5020/…
– user401938
Jul 19 at 21:39












"trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
– DJohnM
Jul 19 at 21:43





"trigonometric trick is required to show RHS"... So start over with the RHS and reduce it to sin and cos...
– DJohnM
Jul 19 at 21:43











2 Answers
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up vote
3
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accepted










$$sec^2 A + csc^2 A = frac 1cos^2 A + frac 1sin^2 A= frac sin^2 A +cos^2 Asin^2 A cos^2 A$$



Note that $$sin A cos A = (1/2)sin 2A $$



Therefore $$frac sin^2 A +cos^2 Asin^2 A cos^2 A=frac 1sin^2 A cos^2 A = 4csc^2 2A$$






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  • Great, I get it, thank you!
    – number8
    Jul 19 at 21:48










  • @number8 thanks for your attention.
    – Mohammad Riazi-Kermani
    Jul 19 at 21:53

















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1
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$$frac1cos^2A+frac1sin^2A=frac1cos^2Asin^2A$$
Next use the duplication formula for the sine.






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    2 Answers
    2






    active

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    2 Answers
    2






    active

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    active

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    up vote
    3
    down vote



    accepted










    $$sec^2 A + csc^2 A = frac 1cos^2 A + frac 1sin^2 A= frac sin^2 A +cos^2 Asin^2 A cos^2 A$$



    Note that $$sin A cos A = (1/2)sin 2A $$



    Therefore $$frac sin^2 A +cos^2 Asin^2 A cos^2 A=frac 1sin^2 A cos^2 A = 4csc^2 2A$$






    share|cite|improve this answer























    • Great, I get it, thank you!
      – number8
      Jul 19 at 21:48










    • @number8 thanks for your attention.
      – Mohammad Riazi-Kermani
      Jul 19 at 21:53














    up vote
    3
    down vote



    accepted










    $$sec^2 A + csc^2 A = frac 1cos^2 A + frac 1sin^2 A= frac sin^2 A +cos^2 Asin^2 A cos^2 A$$



    Note that $$sin A cos A = (1/2)sin 2A $$



    Therefore $$frac sin^2 A +cos^2 Asin^2 A cos^2 A=frac 1sin^2 A cos^2 A = 4csc^2 2A$$






    share|cite|improve this answer























    • Great, I get it, thank you!
      – number8
      Jul 19 at 21:48










    • @number8 thanks for your attention.
      – Mohammad Riazi-Kermani
      Jul 19 at 21:53












    up vote
    3
    down vote



    accepted







    up vote
    3
    down vote



    accepted






    $$sec^2 A + csc^2 A = frac 1cos^2 A + frac 1sin^2 A= frac sin^2 A +cos^2 Asin^2 A cos^2 A$$



    Note that $$sin A cos A = (1/2)sin 2A $$



    Therefore $$frac sin^2 A +cos^2 Asin^2 A cos^2 A=frac 1sin^2 A cos^2 A = 4csc^2 2A$$






    share|cite|improve this answer















    $$sec^2 A + csc^2 A = frac 1cos^2 A + frac 1sin^2 A= frac sin^2 A +cos^2 Asin^2 A cos^2 A$$



    Note that $$sin A cos A = (1/2)sin 2A $$



    Therefore $$frac sin^2 A +cos^2 Asin^2 A cos^2 A=frac 1sin^2 A cos^2 A = 4csc^2 2A$$







    share|cite|improve this answer















    share|cite|improve this answer



    share|cite|improve this answer








    edited Jul 19 at 21:58









    amWhy

    189k25219431




    189k25219431











    answered Jul 19 at 21:46









    Mohammad Riazi-Kermani

    27.5k41852




    27.5k41852











    • Great, I get it, thank you!
      – number8
      Jul 19 at 21:48










    • @number8 thanks for your attention.
      – Mohammad Riazi-Kermani
      Jul 19 at 21:53
















    • Great, I get it, thank you!
      – number8
      Jul 19 at 21:48










    • @number8 thanks for your attention.
      – Mohammad Riazi-Kermani
      Jul 19 at 21:53















    Great, I get it, thank you!
    – number8
    Jul 19 at 21:48




    Great, I get it, thank you!
    – number8
    Jul 19 at 21:48












    @number8 thanks for your attention.
    – Mohammad Riazi-Kermani
    Jul 19 at 21:53




    @number8 thanks for your attention.
    – Mohammad Riazi-Kermani
    Jul 19 at 21:53










    up vote
    1
    down vote













    $$frac1cos^2A+frac1sin^2A=frac1cos^2Asin^2A$$
    Next use the duplication formula for the sine.






    share|cite|improve this answer

























      up vote
      1
      down vote













      $$frac1cos^2A+frac1sin^2A=frac1cos^2Asin^2A$$
      Next use the duplication formula for the sine.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        $$frac1cos^2A+frac1sin^2A=frac1cos^2Asin^2A$$
        Next use the duplication formula for the sine.






        share|cite|improve this answer













        $$frac1cos^2A+frac1sin^2A=frac1cos^2Asin^2A$$
        Next use the duplication formula for the sine.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 19 at 21:40









        Bernard

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