If 3 cot A = 4, check whether (1-tan²A)/(1+tan²A) = cos² A – sin² A or not.
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Solution:
Let us consider a triangle ABC, right-angled at B.
As per the given question,
cot A = 4/3 = AB/BC
So, AB = 4k and BC = 3k, where k is any positive integer.
Using Pythagoras theorem,
AC² = AB² + BC²
Putting the values of AB and BC, we get;
AC² = (4k)² + (3k)²
AC² = 16k² + 9k²
AC = √25k² = 5k
Now we have all the sides.
As we know,
tan A = 1/cot A
tan A = 3/4
sin A = BC/AC = 3/5
cos A = AB/AC = 4/5
To check: (1-tan²A)/(1+tan²A) = cos² A – sin² A or not
Let us take L.H.S. first;
(1-tan²A)/(1+tan²A) = 1-(3/4)²/1+(3/4)²
= [1-9/16]/[1+9/16] = 7/25
R.H.S. = cos² A – sin² A = (4/5)²-(3/5)²
= (16/25) – (9/25) = 7/25
Since,
L.H.S. = R.H.S.
.Hence, proved.
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