Having trouble with this:
Verify:
csc^4x-cot^4x=2csc^2x-1
*the numbers after the carets are to the power of
To verify the given trigonometric identity, we need to simplify both sides and show that they are equal to each other. Let's start by simplifying each side of the equation separately.
Starting with the left side:
csc^4(x) - cot^4(x)
Using the trigonometric identities:
csc^2(x) = 1 + cot^2(x)
cot^2(x) = csc^2(x) - 1
We can rewrite the expression as:
(csc^2(x))^2 - ((csc^2(x) - 1))^2
Expanding the squares, we have:
(csc^2(x))^2 - (csc^4(x) - 2csc^2(x) + 1)
Now, simplifying further:
csc^4(x) - 2csc^2(x) + 1 - csc^4(x) + 2csc^2(x) - 1
The -2csc^2(x) and the +2csc^2(x) cancel each other out, leaving us with:
0
Therefore, the left side simplifies to 0.
Moving on to the right side of the equation:
2csc^2(x) - 1
This is already simplified.
Now, comparing the simplified left side (0) to the right side (2csc^2(x) - 1), we can see that they are not equal.
Hence, by simplifying both sides of the equation, we can determine whether the given equation is true or false. In this case, the equation is false because the left side simplifies to 0, but the right side is not equal to 0.