In a triangle prove that sin 2 ⁡ ( A 2 ) + sin 2...

Wronsonia8g

Wronsonia8g

Answered

2022-07-09

In a triangle prove that sin 2 ( A 2 ) + sin 2 ( B 2 ) + sin 2 ( C 2 ) + 2 sin ( A 2 ) sin ( B 2 ) sin ( C 2 ) = 1

Answer & Explanation

diamondogsaz

diamondogsaz

Expert

2022-07-10Added 12 answers

Let x = A 2 , y = B 2 , z = C 2 , x + y + z = π / 2
sin 2 ( x ) + sin 2 ( y ) + sin 2 ( z ) + 2 sin ( x ) sin ( y ) sin ( z ) = sin 2 ( x ) + sin 2 ( y ) + sin 2 ( π / 2 x y ) + 2 sin ( x ) sin ( y ) sin ( π / 2 x y ) = sin 2 ( x ) + sin 2 ( y ) + cos 2 ( x + y ) + 2 sin ( x ) sin ( y ) cos ( x + y ) = sin 2 ( x ) + sin 2 ( y ) + cos 2 ( x + y ) + ( cos ( x y ) cos ( x + y ) ) cos ( x + y ) = sin 2 ( x ) + sin 2 ( y ) + cos ( x y ) cos ( x + y ) = 1 cos ( 2 x ) 2 + 1 cos ( 2 y ) 2 + cos ( x y ) cos ( x + y ) = 1 cos ( 2 x ) + cos ( 2 y ) 2 + cos ( x y ) cos ( x + y ) = 1 cos ( x y ) cos ( x + y ) + cos ( x y ) cos ( x + y ) = 1

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