# If 3 cot A = 4, check whether (1 - tan^{2 }A) / (1 + tan^{2} A) = cos^{2} A - sin^{2} A or not

**Solution:**

We use the basic concepts of trigonometric ratios like cot, tan, cos and sin to solve the question.

Using 3cot A = 4, we can find the ratio of the length of two sides of the right-angled triangle. Then by using the Pythagoras theorem, the third side and required trigonometric ratios.

3 cot A = 4

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.

AC^{2} = AB^{2} + BC^{2}

= (4k)^{2} + (3k)^{2}

= 16k^{2} + 9k^{2}

= 25k^{2}

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 - tan^{2} A) / (1 + tan^{2} A)

= [1 - (3/4)^{2}] / [(1 + (3/4)^{2})]

= (1 - 9/16) / (1 + 9/16)

= (16 - 9) / (16 + 9)

= 7/25

R.H.S = cos^{2 }A - sin^{2} A

(4/5)^{2} - (3/5)^{2}

= 16/25 - 9/25

= (16 - 9)/25

= 7/25

Therefore, (1 - tan^{2} A) / (1 + tan^{2} A) = cos^{2} A - sin^{2} A

**Video Solution:**

## If 3 cot A = 4, check whether (1 - tan^{2} A) / (1 + tan^{2} A) = cos^{2} A - sin^{2} A or not

### Maths NCERT Solutions Class 10 - Chapter 8 Exercise 8.1 Question 8:

If 3 cot A = 4, check whether (1 - tan^{2} A) / (1 + tan^{2} A) = cos^{2} A - sin^{2} A or not

If 3 cot A = 4, then it is true that (1 - tan^{2} A) / (1 + tan^{2} A) = cos^{2} A - sin^{2} A