(cos^4x -sin ^4 x)/(sin^2x)=cot^2x -1
Prove true
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http://www.symbolab.com/solver/trigonometric-identity-calculator/%5Cfrac%7B(cos%5E%7B4%7D(x)-sin%5E%7B4%7D(x))%7D%7Bsin%5E%7B2%7D(x)%7D%3Dcot%5E%7B2%7D(x)-1/?origin=button
To prove that the equation `(cos^4x - sin^4x) / sin^2x = cot^2x - 1` is true, we will start by simplifying the left-hand side of the equation and then simplify the right-hand side of the equation separately. If both sides simplify to the same expression, then the equation is true.
Let's begin:
1. Start with the left-hand side of the equation:
`(cos^4x - sin^4x) / sin^2x`
2. We can rewrite the numerator using the identity `a^2 - b^2 = (a + b)(a - b)`:
`[(cos^2x)^2 - (sin^2x)^2] / sin^2x`
`[(cos^2x + sin^2x)(cos^2x - sin^2x)] / sin^2x`
3. Use the Pythagorean identity `cos^2x + sin^2x = 1` to simplify further:
`(1)(cos^2x - sin^2x) / sin^2x`
`(cos^2x - sin^2x) / sin^2x`
4. Use the identity `cos^2x - sin^2x = cos(2x)` to simplify it further:
`cos(2x) / sin^2x`
5. Rewrite `cos(2x)` as `1 - 2sin^2x` using the double angle formula:
`(1 - 2sin^2x) / sin^2x`
6. Rearrange the numerator:
`1 / sin^2x - 2sin^2x / sin^2x`
7. Simplify:
`csc^2x - 2`
Now, let's simplify the right-hand side of the equation:
1. Start with the right-hand side of the equation:
`cot^2x - 1`
2. Rewrite `cot^2x` in terms of sine and cosine:
`(cos^2x / sin^2x) - 1`
3. Find a common denominator:
`(cos^2x - sin^2x) / sin^2x`
4. Use the identity `cos^2x - sin^2x = cos(2x)` again:
`cos(2x) / sin^2x`
5. Rewrite `cos(2x)` as `1 - 2sin^2x` using the double angle formula:
`(1 - 2sin^2x) / sin^2x`
6. Simplify:
`1 / sin^2x - 2sin^2x / sin^2x`
7. Simplify further:
`csc^2x - 2`
Since the left-hand side `(cos^4x - sin^4x) / sin^2x` simplifies to `csc^2x - 2`, and the right-hand side `cot^2x - 1` also simplifies to `csc^2x - 2`, we can conclude that both sides of the equation are equal. Therefore, the equation `(cos^4x - sin^4x) / sin^2x = cot^2x - 1` is true.