Proof that :- cot²A/(cosec A + 1)=1-sin A/1+sin A.
To prove the given identity, let's manipulate each side of the equation separately and then equate them.
Starting with the left side of the equation:
cot²A / (cosec A + 1)
To work with the left-hand side (LHS), we need to express cot A and cosec A in terms of sin A and cos A using the fundamental trigonometric identities:
1. cot A = cos A / sin A
2. cosec A = 1 / sin A
Substituting these values into the LHS:
(cot² A) / (cosec A + 1)
= (cos² A / sin² A) / (1 / sin A + 1)
= (cos² A / sin² A) / (1 / sin A + sin A / sin A)
= (cos² A / sin² A) / ((1 + sin A) / sin A)
= cos² A / sin² A * sin A / (1 + sin A)
= (cos² A * sin A) / (sin² A * (1 + sin A))
Simplifying further, we get:
(cos² A * sin A) / (sin² A * (1 + sin A))
= [(cos A * sin A) * cos A] / [(sin A * sin A) * (1 + sin A)]
= (cos A * sin A * cos A) / (sin A * sin³ A * (1 + sin A))
= (cos A * cos A) / (sin A * sin² A * (1 + sin A))
= cos² A / (sin² A * (1 + sin A))
Now, let's work on the right side of the equation:
1 - sin A / (1 + sin A)
Combining the fractions, we get:
[(1 * (1 + sin A) - sin A)] / (1 + sin A)
= [1 + sin A - sin A] / (1 + sin A)
= 1 / (1 + sin A)
So, the right-hand side (RHS) becomes:
1 / (1 + sin A)
Now, we need to prove that the LHS = RHS, which means we need to show that:
(cos² A) / (sin² A * (1 + sin A)) = 1 / (1 + sin A)
To prove this, we can start by cross-multiplying:
(cos² A) * (1 + sin A) = sin² A * (1 + sin A) * 1
Expanding both sides:
cos² A + cos² A * sin A = sin² A + sin³ A
Moving all terms to one side:
sin³ A + cos² A * sin A - sin² A - cos² A = 0
Factoring out sin A from the first two terms:
sin A * (sin² A + cos² A) + cos² A * sin A - sin² A - cos² A = 0
Using the identity sin² A + cos² A = 1:
sin A * 1 + cos² A * sin A - sin² A - cos² A = 0
Simplifying:
sin A + cos² A * sin A - sin² A - cos² A = 0
Pulling out common factors:
sin A * (1 + cos² A) - (1 + cos² A) = 0
(1 + cos² A) is a common factor:
(1 + cos² A)(sin A - 1) = 0
Therefore, either (1 + cos² A) = 0 or (sin A - 1) = 0.
Since (1 + cos² A) cannot be zero for any real values of A, we can disregard that solution.
Hence, we are left with sin A - 1 = 0, which simplifies to sin A = 1.
However, sin A cannot be equal to 1 for any real values of A.
Therefore, the equation (cos² A) / (sin² A * (1 + sin A)) = 1 / (1 + sin A) is not a valid identity.
Hence, the original statement is not true.