answer this trigonometric problem please.
Answers
Answer:
x = ( a + b ) / ( 1 - ab )
Step-by-step explanation:
Let A = arctan a. Then...
a = tan A => 1 + a² = sec² A
=> cos² A = 1 / ( 1 + a² )
and sin² A = 1 - cos² A = a² / ( 1 + a² ).
So...
sin² 2A = ( 2 sin A cos A )² = 4 sin²A cos²A = 4a² / ( 1 + a² )²
=> sin 2A = 2a / ( 1 + a² )
=> 2A = arcsin [ 2a / ( 1 + a² ) ]
Similarly, letting B = arctan b, we get 2B = arcsin [ 2b / ( 1 + b² ) ].
Therefore the equation becomes:
2A + 2B = 2 arctan x
=> arctan x = A + B
=> x = tan ( A + B )
= ( tan A + tan B ) / ( 1 - tan A tan B )
=> x = ( a + b ) / ( 1 - ab )
[ N.B. Actually, we haven't been careful about details like the signs of a and b here, so more work is needed really to ensure that this is spot on! But for elementary purposes, this should be adequate. ]