Doing some test corrections and not seeing this....Please help:
Verify.
csc^4x - cot^4x = 2csc^2x-1
To verify the given equation csc^4x - cot^4x = 2csc^2x - 1, we will simplify both sides step-by-step:
Starting with the left side: csc^4x - cot^4x
Step 1: Rewrite csc^4x as (1/sin^2x)^2
= (1/sin^2x)^2 - cot^4x
Step 2: Expand the numerator
= (1^2)/(sin^2x)^2 - cot^4x
= 1/(sin^2x)^2 - cot^4x
Step 3: Simplify the denominator
= 1/sin^4x - cot^4x
Step 4: Rewrite cot^4x as (cos^2x)/(sin^2x)^2
= 1/sin^4x - (cos^2x)/(sin^2x)^2
Step 5: Simplify the fraction with a common denominator
= (1 - cos^2x)/(sin^2x)^2
Step 6: Recall the trigonometric identity sin^2x + cos^2x = 1
= sin^2x/(sin^2x)^2
Step 7: Reduce the exponent
= sin^2x/sin^4x
Step 8: Simplify further
= 1/sin^2x
Now, let's simplify the right side: 2csc^2x - 1
Step 1: Rewrite csc^2x as (1/sin^2x)
= 2(1/sin^2x) - 1
Step 2: Find a common denominator
= (2 - sin^2x)/(sin^2x)
Step 3: Recall the trigonometric identity sin^2x + cos^2x = 1
= (2 - cos^2x)/(sin^2x)
Step 4: Simplify further
= 2/sin^2x - cos^2x/sin^2x
Step 5: Recall that cot^2x = cos^2x/sin^2x
= 2/sin^2x - cot^2x
Now, comparing the result from the left side (1/sin^2x) and the right side (2/sin^2x - cot^2x), we can see that they are not equal. Therefore, the original equation csc^4x - cot^4x = 2csc^2x - 1 is incorrect.
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°.
Not a surprise that you don't get it,
the two expressions are not equivalent!
Start from left:
csc^4(x)-cot^4(x)
Factor sin^4(x) as denominator:
=(1-cos^4(x))/sin^4(x)
Factor as difference of two squares
= (1 + cos²(x)) (1 - cos²(x)) / sin^4(x)
=(1 + cos²(x))sin²(x)/sin^4(x)
=(1+cos²(x))/sin²(x)
=(2cos²(x)+sin²(x))/sin²(x)
=2cos²(x)/sin²(x)+1
=2cot²(x)+1
(and not 2csc²(x)+1)