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Grade: 12

                        

If 3 cot A = 4, check whether (1-tan2 A)/(1+tan2 A) = cos2 A – sin 2 A or not.

one month ago

Answers : (1)

Anand Kumar Pandey
askIITians Faculty
1492 Points
							Dear Student
the question should be
1-tan^2(A)/1+tan^2(A)= cos^2-sin^2A

Let△ABC in which∠B=90°
We know that, cot function is the reciprocal of tan function and it is written ascot(A) = AB/BC = 4/3
Let AB = 4k an BC =3k, where k is a positive real number.
According to the Pythagorean theorem,
AC^2=AB^2+BC^2
AC^2=(4k)^2+(3k)^2
AC^2=16k^2+9k^2
AC^2=25k^2
AC=5k
Now, apply the values corresponding to the ratios

tan(A) = BC/AB = 3/4
sin (A) = BC/AC = 3/5
cos (A) = AB/AC = 4/5
Now compare the left hand side(LHS) with right hand side(RHS)

put the values in LHS and RHS
LHS= (1-tan^2A)/(1+tan^2A)=7/25
RHS=cos^2A–sin^2A= 7/25
Since, both the LHS and RHS = 7/25
R.H.S. =L.H.S.

Hence,(1-tan^2A)/(1+tan^2A) = cos^2A–sin^2Ais proved

Thanks

one month ago
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