Are you smarter than me? 14

Geometry Level 5

If tan ⁡ ( A ) . s e c ( 4 A ) + t a n ( 4 A ) = t a n ( x ) + t a n ( 4 A ) . s e c ( y ) \tan { (A) } .sec(4A)+tan(4A)=tan(x)+tan(4A).sec(y)

And A = 10000 A=10000 Find x + y x+y


The answer is 30000.

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Mehul Chaturvedi by Mehul Chaturvedi , 22, India

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1 solution

Mehul Chaturvedi
Dec 5, 2014

W e n e e d t o p r o v e t h a t t a n A . s e c 4 A + t a n 4 A = t a n A + t a n 4 A . s e c 2 A T h e p r o b l e m C a n b e r e w r i t t e n a s : t a n A . s e c 4 A − t a n A = t a n 4 A . s e c 2 A − t a n 4 A . t a n A ( s e c 4 A − 1 ) = t a n 4 A ( s e c 2 A − 1 ) ⇒ s e c 4 A − 1 s e c 2 A − 1 = t a n 4 A t a n A . . . . . . E q n . 1 s e c 4 A − 1 s e c 2 A − 1 = 1 c o s 4 A − 1 1 c o s 2 A − 1 L H S = = 1 − c o s 4 A c o s 4 A . c o s 2 A 1 − c o s 2 A = 2 s i n 2 2 A c o s 4 A . c o s 2 A 2 s i n 2 A = 2 s i n 2 A . c o s 2 A . s i n 2 A c o s 4 A . 2 s i n 2 A = s i n 4 A . 2 s i n A c o s A c o s 4 A . 2 s i n 2 A = t a n 4 A . c o t A = t a n 4 A t a n A f r o m E q n . 1 R H S = t a n 4 A t a n A h e n c e L H S = R H S H e n c e p r o v e d H e n c e x + y = 3 A w h e r e A = 10000 t h e r e f o r e 3 A = 30000 We\quad need\quad to\quad prove\quad that\\ tanA.sec4A+tan4A=tanA+tan4A.sec2A\\ The\quad problem\quad Can\quad be\quad rewritten\quad as:\\ \quad \quad tanA.sec4A-tanA=tan4A.sec2A\\ -tan4A.tanA(sec4A-1)=tan4A(sec2A-1)\\ \Rightarrow \frac { sec4A-1 }{ sec2A-1 } =\frac { tan4A }{ tanA } ......Eqn.1\\ \frac { sec4A-1 }{ sec2A-1 } =\frac { \frac { 1 }{ cos4A } -1 }{ \frac { 1 }{ cos2A } -1 } \\ LHS=\\ =\frac { 1-cos4A }{ cos4A } .\frac { cos2A }{ 1-cos2A } \\ =\frac { 2{ sin }^{ 2 }2A }{ cos4A } .\frac { cos2A }{ { 2sin }^{ 2 }A } \\ =\frac { 2sin2A.cos2A.sin2A }{ cos4A.2{ sin }^{ 2 }A } \\ =\frac { sin4A.2sinAcosA }{ cos4A.2{ sin }^{ 2 }A } \\ =tan4A.cotA\\ =\frac { tan4A }{ tanA } \\ from\quad Eqn.\quad 1\quad RHS=\frac { tan4A }{ tanA } \\ hence\quad LHS=RHS\quad \\ Hence\quad proved\\ Hence\quad x+y=3A\\ where\quad A=10000\quad therefore\quad \\ 3A=30000

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What made you assume x=A and y=2A? Is there another way of getting to the answer without knowing it in the first place? Also the 5th line has an extra term tan(4A).

Dhruva Patil - 6 years, 5 months ago

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