To verify the given equation: csc^4x - cot^4x = 2csc^2x - 1, we can use trigonometric identities to simplify both sides of the equation and see if they are equal.
First, let's recall the basic trigonometric identities:
1. csc^2x = 1/sin^2x
2. cot^2x = cos^2x/sin^2x
Now, let's start by simplifying the left side of the equation:
csc^4x - cot^4x
Replacing csc^2x and cot^2x with their identities:
(1/sin^2x)^2 - (cos^2x/sin^2x)^2
Simplifying further:
(1/sin^2x)^2 - (cos^2x)^2/sin^4x
Expanding the square:
(1/sin^4x) - (cos^4x/sin^4x)
Finding a common denominator:
(1 - cos^4x)/sin^4x
Now, let's simplify the right side of the equation:
2csc^2x - 1
Replacing csc^2x with its identity:
2(1/sin^2x) - 1
Finding a common denominator:
(2 - sin^2x)/sin^2x
Now we can compare both sides of the equation:
(1 - cos^4x)/sin^4x = (2 - sin^2x)/sin^2x
To check if the equation is true, we can cross-multiply and simplify:
sin^2x(1 - cos^4x) = sin^2x(2 - sin^2x)
Expanding both sides:
sin^2x - sin^2x*cos^4x = 2sin^2x - sin^4x
Rearranging terms:
sin^4x - sin^2x = 0
Factoring out sin^2x:
sin^2x(sin^2x - 1) = 0
Now, we have two possible solutions:
1. sin^2x = 0
2. sin^2x - 1 = 0
For sin^2x = 0, we have:
sin^2x = 0
sinx = 0
This gives us the solution x = 0° and x = 180°.
For sin^2x - 1 = 0, we have:
sin^2x - 1 = 0
sin^2x = 1
sinx = ±1
This gives us the solutions x = 90° and x = 270°.
Therefore, the equation csc^4x - cot^4x = 2csc^2x - 1 is verified to be true for x = 0°, 90°, 180°, and 270°.