Prove: 1 + (

delirija7z 2022-07-09 Answered
Prove:
1 + ( tan x sin y ) 2 1 + ( tan x sin z ) 2 = 1 + ( sin x tan y ) 2 1 + ( sin x tan z ) 2
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Answers (2)

Zichetti4b
Answered 2022-07-10 Author has 13 answers
Notice, the given equality can be easily proved by simplifying LHS
L H S = 1 + ( tan x sin y ) 2 1 + ( tan x sin z ) 2
= 1 + ( sin x sin y cos x ) 2 1 + ( sin x sin z cos x ) 2
= sin 2 z ( sin 2 y cos 2 x + sin 2 x ) sin 2 y ( sin 2 z cos 2 x + sin 2 x )
= sin 2 z ( ( 1 cos 2 y ) ( 1 sin 2 x ) + sin 2 x ) sin 2 y ( ( 1 cos 2 z ) ( 1 sin 2 x ) + sin 2 x )
= sin 2 z ( 1 cos 2 y sin 2 x + sin 2 x cos 2 y + sin 2 x ) sin 2 y ( 1 cos 2 z sin 2 x + sin 2 x cos 2 z + sin 2 x )
= sin 2 z ( ( 1 cos 2 y ) + sin 2 x cos 2 y ) sin 2 y ( ( 1 cos 2 z ) + sin 2 x cos 2 z )
= sin 2 z ( sin 2 y + sin 2 x cos 2 y ) sin 2 y ( sin 2 z + sin 2 x cos 2 z )
= sin 2 z sin 2 y ( 1 + sin 2 x cos 2 y sin 2 y ) sin 2 y sin 2 z ( 1 + sin 2 x cos 2 z sin 2 z )
= 1 + sin 2 x tan 2 y 1 + sin 2 x tan 2 z
= 1 + ( sin x tan y ) 2 1 + ( sin x tan z ) 2 = R H S

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Montenovofe
Answered 2022-07-11 Author has 3 answers
1 + ( tan x sin y ) 2 1 + ( tan x sin z ) 2 = 1 + ( sin x tan y ) 2 1 + ( sin x tan z ) 2
1 + ( tan x sin y ) 2 1 + ( sin x tan y ) 2 = 1 + ( tan x sin z ) 2 1 + ( sin x tan z ) 2
So, if we can prove that 1 + ( tan x sin A ) 2 1 + ( sin x tan A ) 2 is independent of A, we are done.
Method #1:
1 + ( tan x sin A ) 2 1 + ( sin x tan A ) 2 = 1 + tan 2 x csc 2 A 1 + sin 2 x cot 2 A = cos 2 A + sin 2 x ( 1 + cot 2 A ) cos 2 x ( 1 + sin 2 x cot 2 A ) = sec 2 x which is clearly independent of A
Method #2:
1 + ( tan x sin A ) 2 1 + ( sin x tan A ) 2 = ( sin 2 A cos 2 x + sin 2 x ) sin 2 A sin 2 A cos 2 x ( sin 2 A + cos 2 A sin 2 x ) = sec 2 x as sin 2 A cos 2 x + sin 2 x = sin 2 A ( 1 sin 2 x ) + sin 2 x = sin 2 A + sin 2 x ( 1 sin 2 A ) = ?

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