Ex 8.4, 5 (iii) - Chapter 8 Class 10 Introduction to Trignometry (Term 1)

Transcript

Ex 8.4, 5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii)tan /( 1 cot " " )+cot /(1 tan ) =1+ sec cosec [Hint : Write the expression in terms of sin and cos ] Taking L.H.S tan /( 1 cot " " )+cot /(1 tan ) = (sin /cos )/(1 (cos /sin ) )+((cos /sin ))/(1 (sin /cos ) ) = (sin /cos )/(((sin cos )/sin ) )+((cos /sin ))/( ((cos sin )/cos ) ) = sin /cos sin /(sin cos ) +cos /sin cos /(cos sin ) = sin2 /( cos ( sin cos ) ) + cos2 /( sin (cos sin ) ) = sin2 /( cos ( sin cos ) ) + cos2 /( sin ( sin cos ) ) = sin2 /( cos (sin cos ) ) cos2 /( sin ( sin cos ) ) = (sin2 (sin ) cos2 (cos ))/( cos ( sin cos )sin ) = (sin3 cos3 )/( cos sin ( sin cos ) ) = ( (sin cos ) (sin2 + cos2 +cos sin ))/( cos sin ( sin cos ) ) = ( (sin2 + cos2 +cos sin ))/(cos sin ) = ( (sin2 + cos2 ) + cos sin )/(cos sin ) = (1 + cos sin )/(cos sin ) = (1 )/(cos sin ) + (cos sin )/(cos sin ) = (1 )/cos (1 )/sin + 1 = sec cosec + 1 = 1 + sec cosec = R.H.S , L.H.S = R.H.S Hence proved