If 3 cot A = 4, check whether (1 - tan2 A) / (1 + tan2 A) = cos2 A - sin2 A or not
Solution:
We use the basic concepts of trigonometric ratios like cot, tan, cos, and sin to solve the question.
3 cot A = 4
Thus, cot A = 4/3
Let ΔABC be a right-angled triangle where angle B is a right angle.
cot A = side adjacent to ∠A / side opposite to ∠A = AB/BC = 4/3
Let AB = 4k and BC = 3k, where k is a positive integer.
By applying the Pythagoras theorem in ΔABC, we get,
AC2 = AB2 + BC2
= (4k)2 + (3k)2
= 16k2 + 9k2
= 25k2
AC = √25k²
= 5k
Therefore,
tan A = side opposite to ∠A / side adjacent to ∠A = BC/AB = 3k/4k = 3/4
sin A = side opposite to ∠A / hypotenuse = BC/AC = 3k/5k = 3/5
cos A = side adjacent to ∠A / hypotenuse = AB/AC = 4k/5k = 4/5
L.H.S = (1 - tan2 A) / (1 + tan2 A)
= [1 - (3/4)2] / [1 + (3/4)2]
= (1 - 9/16) / (1 + 9/16)
= (16 - 9) / (16 + 9)
= 7/25
R.H.S = cos2 A - sin2 A
(4/5)2 - (3/5)2
= 16/25 - 9/25
= (16 - 9) / 25
= 7/25
Therefore, (1 - tan2 A) / (1 + tan2 A) = cos2 A - sin2 A
☛ Check: NCERT Solutions for Class 10 Maths Chapter 8
Video Solution:
If 3 cot A = 4, check whether (1 - tan² A) / (1 + tan² A) = cos² A - sin² A or not.
Maths NCERT Solutions Class 10 Chapter 8 Exercise 8.1 Question 8
Summary:
If 3 cot A = 4, then it is true that (1 - tan2 A) / (1 + tan2 A) = cos2 A - sin2 A.
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