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If 3 cot A = 4, check whether (1-tan2 A)/(1+tan2 A) = cos2 A – sin 2 A or not.
Dear Studentthe question should be1-tan^2(A)/1+tan^2(A)= cos^2-sin^2ALet△ABC in which∠B=90°We know that, cot function is the reciprocal of tan function and it is written ascot(A) = AB/BC = 4/3Let AB = 4k an BC =3k, where k is a positive real number.According to the Pythagorean theorem,AC^2=AB^2+BC^2AC^2=(4k)^2+(3k)^2AC^2=16k^2+9k^2AC^2=25k^2AC=5kNow, apply the values corresponding to the ratiostan(A) = BC/AB = 3/4sin (A) = BC/AC = 3/5cos (A) = AB/AC = 4/5Now compare the left hand side(LHS) with right hand side(RHS)put the values in LHS and RHSLHS= (1-tan^2A)/(1+tan^2A)=7/25RHS=cos^2A–sin^2A= 7/25Since, both the LHS and RHS = 7/25R.H.S. =L.H.S.Hence,(1-tan^2A)/(1+tan^2A) = cos^2A–sin^2Ais provedThanks
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