Example 5 - Chapter 2 Class 12 Inverse Trigonometric Functions

Transcript

Example 5 Express tan-1 cos x/(1 sin x ) , - /2 < x < 3 /2 in the simplest form Lets first calculate cos x & 1 sin x Solving tan-1 (cos x/(1 sin x )) = tan-1 [(cos2 x/2 sin2 x/2)/(1 (2 sin x/2 cos x/2 ) )] = tan-1 [(cos2 x/2 sin2 x/2)/(1 2 sin x/2 cos x/2 )] = tan-1 [(cos2 x/2 sin2 x/2)/(cos2 x/2 + sin2 x/2 2 sin x/2 cos x/2 )] = tan-1 [(cos x/2 + sin x/2)(cos x/2 sin x/2)/(cos x/2 sin x/2)^2 ] = tan-1 [((cos x/2 + sin x/2))/((cos x/2 sin x/2) )] Dividing by cos /2 = tan-1 ((cos /( 2 ) / /2 + sin /( 2 ) / /2 )/( /( 2 ) / /2 /( 2 ) / /2 )) = tan-1 [(1 + tan /2 )/(1 tan /2 )] = tan-1 [(1 + tan /2 )/(1 1. tan /2 )] = tan-1 (( /4 + /2 )/( 1 /4 . /2 )) = tan -1 [tan ( /4+x/2 ) ] = /4+x/2