What is sin^2 - cos^2 x/ 1-2cos^2 x, using the identities.
cos 2A
= cos^2 A - sin^2 A
or 2cos^2 A - 1
or 1 - 2sin^2 A
so (sin^2 - cos^2 x)/ (1-2cos^2 x)
= -cos 2x / - cos 2x
= +1
To simplify the given expression using trigonometric identities, we can start by factoring the numerator and denominator expressions using the identity:
cos^2 x + sin^2 x = 1
First, let's rewrite the given expression using this identity:
sin^2 x - cos^2 x / 1 - 2cos^2 x
Using the identity mentioned above, we can rewrite the denominator:
(1 - cos^2 x) - cos^2 x
Expanding the numerator using the difference of squares, we get:
(sin x + cos x) * (sin x - cos x) / (1 - cos^2 x) - cos^2 x
Next, we can factor the denominator further using the difference of squares:
(sin x + cos x) * (sin x - cos x) / (1 - cos x)(1 + cos x) - cos^2 x
Simplifying the denominator, we have:
(sin x + cos x) * (sin x - cos x) / (1 - cos^2 x) - cos^2 x
We can rewrite cos^2 x as 1 - sin^2 x:
(sin x + cos x) * (sin x - cos x) / (1 - (1 - sin^2 x)) - (1 - sin^2 x)
Simplifying further, we have:
(sin x + cos x) * (sin x - cos x) / sin^2 x + sin^2 x
Combining like terms in the denominator, we get:
(sin x + cos x) * (sin x - cos x) / 2sin^2 x
Therefore, the simplified form of the expression sin^2 x - cos^2 x / 1 - 2cos^2 x using trigonometric identities is:
(sin x + cos x) * (sin x - cos x) / 2sin^2 x