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 .)

#Algebra #Vectors #Trignometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hint: Write the given equation as

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

$\ddot\smile$

Karthik Kannan - 6 years, 7 months ago

@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 - 6 years, 7 months ago

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 - 6 years, 7 months ago

Please answer my doubt deepanshu gupta

sandeep Rathod - 6 years, 7 months ago

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 - 6 years, 7 months ago

HINT: USE cauchy inequality

Lakshya Kumar - 6 years, 7 months ago

Yes that is best!!

Deepanshu Gupta - 6 years, 7 months ago

could you please explain how?

Aman Gautam - 6 years, 7 months ago

@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 - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$