Question

M tan theta minus 30 is equal to n tan theta + 120

Answer 1

Tan ( x - 120°) = tan [ 90° + ( 30° + x ) ] = - cot ( 30° + x ).  

And tan ( x - 30°) = - tan ( 30° - x ) …………………. (1)  

Now the given relation can be rearranged as ---  

=> m/n = tan ( x + 120°) / tan ( x - 30°),  

=> m/n = - cot ( 30°+ x ) / - tan ( 30° - x )  

=> m/n = [ cos (30° + x ) . cos (30° - x ) ] / [sin (30° + x ) . sin (30° - x )]  

=> (m/n) = ( cos² 30° - sin² x ) / (sin² 30° - sin²x )  

=> (m/n) = ( ¾ - sin² x ) / ( ¼ - sin²x )  

Now apply componendo and dividendo.  

( m + n )/ ( m – n ) = (1 – 2 sin²x) / ( ½ )  

=> ( m + n )/ ( m – n ) = 2 cos 2x ……………………… Proved Proved  

Shaun Abraham

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Answer 2

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