(cos^4x -sin ^4 x)/(sin^2x)=cot^2x -1
To prove the given expression, we need to simplify both sides of the equation and show that they are equal.
Let's start with the left side:
(cos^4x - sin^4x)/(sin^2x)
We can use the identity cos^2x = 1 - sin^2x to rewrite the numerator:
((1-sin^2x)^2 - sin^4x)/(sin^2x)
Expanding the numerator:
((1-2sin^2x+sin^4x) - sin^4x)/(sin^2x)
Cancelling like terms:
(1-2sin^2x)/(sin^2x)
Now, let's simplify the right side of the equation:
cot^2x - 1
Using the identity cot^2x = 1 + tan^2x:
(1 + tan^2x) - 1
Cancelling like terms:
tan^2x
Therefore, we have:
(1-2sin^2x)/(sin^2x) = tan^2x
To prove the equality, we can use the identity sin^2x + cos^2x = 1. Rearranging this identity:
sin^2x = 1 - cos^2x
Substituting sin^2x = 1 - cos^2x into the left side of the equation:
(1-2(1-cos^2x))/(1-cos^2x)
Simplifying:
(1-2+2cos^2x)/(1-cos^2x)
(-1+2cos^2x)/(1-cos^2x)
Using the identity cos^2x = 1 - sin^2x:
(-1+2(1-sin^2x))/(1-(1-sin^2x))
Simplifying:
(-1 + 2 -2sin^2x)/(1-1+sin^2x)
(-2sin^2x + 1)/(sin^2x)
Now, let's simplify the right side of the equation:
tan^2x
Using the identity tan^2x = sin^2x/cos^2x:
sin^2x/cos^2x
Multiplying both numerator and denominator by sin^2x:
sin^4x/(cos^2x * sin^2x)
Using the identity cos^2x = 1 - sin^2x:
sin^4x/((1-sin^2x) * sin^2x)
Expanding the denominator:
sin^4x/(sin^2x - sin^4x)
Using the identity sin^2x = 1 - cos^2x:
sin^4x/((1-cos^2x) - sin^4x)
sin^4x/(1-cos^2x - sin^4x)
Now, let's simplify further:
sin^4x/(1 - cos^2x - sin^4x) = (-2sin^2x + 1)/(sin^2x)
Since the left side of the equation is equal to the right side, we have proved that:
(cos^4x - sin^4x)/(sin^2x) = cot^2x - 1