If 2(\sin a+\cos a)\sin b=3-\cos b, then find 3\tan^2 a+4\tan^2 b I doubt there is a

Rudy Koch

Rudy Koch

Answered question

2022-01-24

If 2(sina+cosa)sinb=3cosb, then find 3tan2a+4tan2b
I doubt there is a solution on the internet now, but they do have a number 35 as the final answer.
My tiny little progress:
Use simple simplification,
3tan2a+4tan2b=3cos2a+4cos2b7
Intuitively, in order to get this, the given equation should be squared. But that is where I got stuck, because the term cosb seems hard to handle; it just seems different from other terms.

Answer & Explanation

Kyler Jacobson

Kyler Jacobson

Beginner2022-01-25Added 8 answers

Rewrite it like this
2(sina+cosa)sinb+cosbE=3
Since we have by Cauchy inequality
sina+cosa(sin2a+cos2a)(12+12)=2
we have always
E8sinb+cosb
Now again by Cauchy inequality
8sinb+cosb3
Since we have equality case we have sinb:cosb=8:1 so tanb=8 . Similary we have tana=1. So the result is 35.

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