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Open Challenge :)

Let \(x,y\quad \in \quad R\) such that \(\cos { x } \cos { y } \quad +\quad \quad 2\sin { y } \quad +\quad 2\sin { x } \cos { y } \quad =\quad 3\).
Then find the value of : \(\tan ^{ 2 }{ x } +\quad 5\tan ^{ 2 }{ y } \quad =\quad ?\).

It is really very beautiful question. That's why I share this with our Brilliant community.

Use any Tool of mathematics. There are no restrictions

(You may use Vectors, Complex numbes, Trigonometry, etc .)

Note by Deepanshu Gupta
2 years, 4 months ago

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Hint: Write the given equation as

\(\bigg(\cos x+2\sin x\bigg)\cos y+2\sin y=3\)

\(\ddot\smile\) Karthik Kannan · 2 years, 4 months ago

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@Karthik Kannan @Karthik Kannan Did you mean this way :

\(E\quad \quad \quad =\quad (\cos x+2\sin x)\cos y+2\sin y\\ \\ { E }_{ max }\quad =\quad \sqrt { { (\cos x+2\sin x) }^{ 2 }+\quad { (2) }^{ 2 } } \quad \\ \\ \quad \quad \quad \quad =\quad \sqrt { \cfrac { 13 }{ 2 } \quad -\quad \cfrac { 3 }{ 2 } \cos { 2x } \quad +\quad 2\sin { 2x } } \\ \quad \quad \quad \quad \\ \quad { E }_{ max }\quad \in \quad \left[ 2\quad ,\quad 3 \right] \\ \\ { E }_{ max }\quad \le \quad 3\\ \\ But\quad \quad \quad { E }_{ max }\quad =\quad 3\\ \).

Now using boundedness of function ! Great ! This is also Nice Method !! Deepanshu Gupta · 2 years, 4 months ago

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HINT: USE cauchy inequality Lakshya Kumar · 2 years, 4 months ago

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@Lakshya Kumar Yes that is best!! Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta could you please explain how? Aman Gautam · 2 years, 4 months ago

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@Aman Gautam Let two Vectors such that :

\[\quad \overset { \rightarrow }{ A } \quad =\quad \cos { x } \cos { y } \quad +\quad \sin { y } +\quad \quad \sin { x } \cos { y } \\ \\ \quad \overset { \rightarrow }{ B } \quad =\quad \overset { \^ }{ i } \quad +\quad 2\overset { \^ }{ j } \quad +\quad 2\overset { \^ }{ k } \quad \].

Now Verify that :

\[\overset { \rightarrow }{ A } .\overset { \rightarrow }{ B } \quad =\quad \left| \overset { \rightarrow }{ A } \right| \left| \overset { \rightarrow }{ B } \right| \].

So angle between these two vectors is zero degree , means they are parallel ,

Now use simple condition of Parallel Vectors and get the Answer ! :)

@Aman Gautam I Hope you Got it ! Deepanshu Gupta · 2 years, 4 months ago

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if x,y belongs to R , then the first expression should be true for all real values of x, but we can see here its true only for some specific Sandeep Rathod · 2 years, 4 months ago

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@Sandeep Rathod Please answer my doubt deepanshu gupta Sandeep Rathod · 2 years, 4 months ago

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@Sandeep Rathod wait be clear , I said : \(x,y\quad \in \quad R\). which means x and y are real numbers. it dosn't mean that that expression is true for every x,y .

And if i say \(\forall \quad x,y\quad \in \quad R\quad \quad \). then it means this expression is true for every x ( which means it is identity ) Deepanshu Gupta · 2 years, 4 months ago

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