cot² A/cosec A + 1)²=1-sin A/1+sin A.
I noticed that you made a correction from your previous post, but It still would not work.
It should be:
cot² A/(cosec A + 1)²= (1-sin A)/(1+sin A)
notice the important brackets.
LS = cot² A/(cosec A + 1)²
= (cos^2 A/sin^2 A)/(1/sina + 1)^2
= (cos^2 A/sin^2 A)((1 + sinA)^2/sin^2 A)
= (cos^2 A/sin^2 A)*(sin^2 A)/(1 + sinA)^2
= cos^2 A/(1+sinA)^2
= (1-sin^2 A)/(1+sinA)^2
= (1-sinA)(1+sinA)/( (1+sinA)(1+sinA) )
= (1 - sinA)/(1 + sinA)
= RS
To simplify the given expression, we'll start by manipulating both the left and right sides separately and then equate them.
Let's simplify the left-hand side (LHS) first:
cot² A / cosec A + 1)²
Using the reciprocal identities,
cot A = 1/tan A and cosec A = 1/sin A
Thus, our expression becomes:
(1/tan A)² / (1/sin A) + 1)²
Simplifying further, we can rewrite (1/sin A) as cosec A, and (1/tan A)² as cot² A:
cot² A / cosec A + 1)²
Now, let's simplify the right-hand side (RHS):
1 - sin A / 1 + sin A
To simplify further, we'll multiply the numerator and denominator of the fraction by (1 - sin A):
(1 - sin A) / (1 + sin A) * (1 - sin A) / (1 - sin A)
Expanding the numerator and denominator:
(1 - 2sin A + sin² A) / (1 - sin² A)
Using the Pythagorean identity (1 - sin² A = cos² A), we can simplify further:
(1 - 2sin A + sin² A) / cos² A
Now, we can rewrite cot² A as (cos² A / sin² A) in order to match the denominator:
(cos² A / sin² A) / cos² A
Simplifying the expression, the RHS becomes:
1/sin² A
Now, we equate the LHS and RHS:
cot² A / cosec A + 1)² = 1 - sin A / 1 + sin A
Substituting the simplified expressions:
1/sin² A = 1 - sin A / 1 + sin A
To solve this equation, we'll cross multiply:
(1/sin² A) * (1 + sin A) = 1 - sin A
Expanding the equation:
1 + sin A = sin² A - sin² A
Since both sides of the equation cancel each other out, we're left with:
1 = 0
However, this makes the equation invalid, as 1 does not equal 0. Therefore, there is no solution to the given equation cot² A / cosec A + 1)² = 1 - sin A / 1 + sin A.