SIMPLIFY SIN RAISE TO POWER 4 MINUS COS RAISE TO POWER4 DIVIDED BY 1 MINUS 2SIN RAISE TO POWER 2
ever heard of math notation?
(sin^4x-cos^4x)/(1-sin^2x)
= (sin^2x+cos^2x)(sin^2x-cos^2x)/cos^2x
= tan^2x-1
To simplify the expression:
(Sin^4(x) - Cos^4(x))/(1 - 2Sin^2(x))
We can start by factoring the numerator and denominator.
The numerator can be written as:
(Sin^2(x))^2 - (Cos^2(x))^2
Using the difference of squares identity, we have:
(Sin^2(x) + Cos^2(x))(Sin^2(x) - Cos^2(x))
Simplifying further, we know that Sin^2(x) + Cos^2(x) is equal to 1 due to the Pythagorean identity. So our numerator can be written as:
1 * (Sin^2(x) - Cos^2(x))
Now let's factor the denominator.
1 - 2Sin^2(x) can be rewritten as:
1 - (2Sin^2(x))
Using the distributive property, we can write it as:
1 - 2(Sin^2(x))
Now we can rewrite the entire expression as:
(Sin^2(x) - Cos^2(x))/(1 - 2Sin^2(x))
We know that Sin^2(x) - Cos^2(x) can be written as -Cos^2(x) + Sin^2(x) by rearranging the terms. So our expression becomes:
(-Cos^2(x) + Sin^2(x))/(1 - 2Sin^2(x))
Finally, we can further simplify by factoring out a negative sign from the numerator:
-(Cos^2(x) - Sin^2(x))/(1 - 2Sin^2(x))
Since Cos^2(x) - Sin^2(x) can be rewritten as -Sin^2(x) + Cos^2(x), our expression simplifies to:
-(Sin^2(x) - Cos^2(x))/(1 - 2Sin^2(x))
And we can rewrite Sin^2(x) - Cos^2(x) as -1 * (Cos^2(x) - Sin^2(x)):
-(-1 * (Cos^2(x) - Sin^2(x)))/(1 - 2Sin^2(x))
Simplifying the double negative sign, we get:
(Cos^2(x) - Sin^2(x))/(1 - 2Sin^2(x))
Therefore, the simplified expression is (Cos^2(x) - Sin^2(x))/(1 - 2Sin^2(x)).