sin^2x-1=sec^2x-tan^2x
To solve the equation sin^2(x) - 1 = sec^2(x) - tan^2(x), we'll need to apply some trigonometric identities.
First, let's rewrite sec^2(x) and tan^2(x) using their equivalent forms in terms of sin(x) and cos(x):
sec^2(x) = (1/cos^2(x))
tan^2(x) = (sin^2(x)/cos^2(x))
Substituting these identities into our equation, we have:
sin^2(x) - 1 = (1/cos^2(x)) - (sin^2(x)/cos^2(x))
To combine the fractions on the right side, we need a common denominator. The common denominator in this case is cos^2(x). Rewriting the equation accordingly:
sin^2(x) - 1 = (1 - sin^2(x))/cos^2(x)
Now, we can multiply both sides of the equation by cos^2(x) to eliminate the denominators:
cos^2(x) * (sin^2(x) - 1) = 1 - sin^2(x)
Expanding the left side of the equation:
sin^2(x) * cos^2(x) - cos^2(x) = 1 - sin^2(x)
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can replace sin^2(x) in the equation:
(1 - cos^2(x)) * cos^2(x) - cos^2(x) = 1 - (1 - cos^2(x))
Simplifying both sides of the equation:
cos^2(x) - cos^4(x) - cos^2(x) = cos^2(x)
Combining like terms on the left side:
- cos^4(x) = 0
Now that we have a quadratic equation in terms of cos(x), we can solve it by factoring.
Factor out a common factor of -cos^2(x):
cos^2(x) * (cos^2(x) - 1) = 0
This equation can be true if either cos^2(x) = 0 or (cos^2(x) - 1) = 0.
For cos^2(x) = 0, the only solution is x = π/2.
For cos^2(x) - 1 = 0, we have cos^2(x) = 1, which has two solutions: x = 0 and x = π.
Therefore, the solutions to the equation sin^2(x) - 1 = sec^2(x) - tan^2(x) are x = 0, x = π/2, and x = π.