verify (csc^4-1)/cot^2x=2+cot^2x
So this is what I have so far on the left side
(csc^2x+1)(cscx+1)(cscx-1)/cot^2x
=(csc^2x+1)(cot^2x)/cot^2x
i think I'm doing something wrong. Please help!
recall that 1 + cot^2 x = csc^2 x
as a variation of the Pythagorean identity
so
LS = (csc^2x+1)(csc^2 x-1)/cot^2x
= (1 + cot^2 x + 1)(1 + cot^2 x -1)/cot^2 x)
= 2 + cot^2 x
= RS
To verify the given equation, let's simplify the left side of the equation step by step.
Starting with the left side of the equation: (csc^4x - 1)/cot^2x
Step 1: Use the identity cot^2x = 1/(tan^2x)
The equation becomes: (csc^4x - 1)/(1/(tan^2x))
Step 2: Apply the identity csc^2x = 1 + cot^2x
The equation becomes: ((1 + cot^2x)^2 - 1)/(1/(tan^2x))
Step 3: Simplify the numerator using algebraic expansion:
To expand (1 + cot^2x)^2, we need to multiply (1 + cot^2x) by itself.
(1 + cot^2x) * (1 + cot^2x) = 1 + 2cot^2x + cot^4x
The equation becomes: ((1 + 2cot^2x + cot^4x) - 1)/(1/(tan^2x))
Step 4: Simplify the numerator by canceling out the 1's:
(2cot^2x + cot^4x)/(1/(tan^2x))
Step 5: Multiply the numerator and denominator by (tan^2x) to simplify further:
[(2cot^2x + cot^4x) * (tan^2x)] / 1
Step 6: Distribute the (tan^2x) in the numerator:
(2cot^2x * tan^2x + cot^4x * tan^2x) / 1
Step 7: Apply the identity cot^2x = 1/(tan^2x):
(2 * (1/(tan^2x)) * tan^2x + cot^4x * tan^2x) / 1
Simplifying gives: (2 + cot^4x * tan^2x) / 1
Step 8: Apply the identity cot^2x = 1/(tan^2x) again:
(2 + (1/(tan^2x))^2 * tan^2x) / 1
Simplifying the square gives: (2 + (1/tan^2x) * tan^2x) / 1
Finally, applying the identity (1/tan^2x) * tan^2x = 1:
(2 + 1) / 1
Which simplifies to: 3
Therefore, the simplified left side of the equation is 3.
Now, let's check the right side of the equation: 2 + cot^2x
Since the right side of the equation is 2 + cot^2x, we can see that it simplifies to the number 2.
Hence, the simplified left side (3) is equal to the right side (2), verifying the given equation (csc^4x - 1)/cot^2x = 2 + cot^2x.
You were on the right track, but made a minor error in Step 3 when expanding the numerator.