solution if x=cosec A + cos A and y=cosec A-cos A then prove that (2/x+y)sqare+(x-y/2)sqare-1=0
x+y = cscA+cosA+cscA-cosA = 2cscA
x-y = cscA+cosA-(cscA-cosA) = 2cosA
2/2cscA = sinA
2cosA/2 = cosA
All clear now?
To prove the given expression, we need to simplify it and show that it equals zero.
Let's start by calculating the values of x and y in terms of A:
x = cosec A + cos A
= 1/sin A + cos A
= (1 + sin A * cos A) / sin A
Similarly,
y = cosec A - cos A
= 1/sin A - cos A
= (1 - sin A * cos A) / sin A
Now, let's substitute these values of x and y into the expression we want to prove:
(2/x + y)^2 + (x - y/2)^2 - 1
First, simplify (2/x + y)^2:
(2/x + y)^2
= (2/(1 + sin A * cos A) + (1 - sin A * cos A) / sin A)^2
To simplify this expression, we need to find a common denominator for the terms. The common denominator will be sin A. So, we multiply the first term by sin A / sin A and the second term by (1 + sin A * cos A) / (1 + sin A * cos A):
= ((2 * sin A) / (sin A * (1 + sin A * cos A))) + ((1 + sin A * cos A) * (1 - sin A * cos A)) / (sin A * (1 + sin A * cos A)))^2
Simplifying further:
= (2 * sin A + (1 + sin A * cos A) * (1 - sin A * cos A)) / (sin A * (1 + sin A * cos A)))^2
= (2 * sin A + (1 - (sin A * cos A)^2)) / (sin A * (1 + sin A * cos A)))^2
= (2 * sin A + (1 - sin^2 A * cos^2 A)) / (sin A * (1 + sin A * cos A)))^2
= (2 * sin A + (1 - sin^2 A * cos^2 A)) / (sin A + sin^2 A * cos A))^2
Next, let's simplify (x - y/2)^2:
(x - y/2)^2
= ((1 + sin A * cos A) / sin A - (1 - sin A * cos A) / (2 * sin A))^2
Again, we need to find a common denominator:
= (((2 * (1 + sin A * cos A)) - (sin A * (1 - sin A * cos A))) / (2 * sin A))^2
Simplifying further:
= (((2 + 2 * sin A * cos A) - sin A + sin^2 A * cos A) / (2 * sin A))^2
= ((1 + 3 * sin^2 A * cos A - sin A) / (2 * sin A))^2
Now, let's substitute these simplified expressions back into the original expression:
(2/x + y)^2 + (x - y/2)^2 - 1
= ( (2 * sin A + (1 - sin^2 A * cos^2 A)) / (sin A + sin^2 A * cos A))^2 + ((1 + 3 * sin^2 A * cos A - sin A) / (2 * sin A))^2 - 1
At this point, it might be challenging to simplify the expression further. You can try expanding the squares and combining like terms, but it's difficult to predict if it will eventually lead to zero. It's possible that there was an error in the initial statement of the problem or further insights are required to prove the given expression.