csc^2x-cos^2xcsc^2x=1
To solve the equation csc^2x - cos^2x csc^2x = 1, we can first simplify the left side of the equation using trigonometric identities, and then solve for x.
Step 1: Simplify the left side of the equation
Since csc^2x is the reciprocal of sin^2x, we can express it as 1/sin^2x.
Similarly, since cos^2x is the square of the cosine function, we can leave it as it is.
Now, substitute these values into the equation:
1/sin^2x - cos^2x / sin^2x = 1
To combine these fractions with a common denominator, multiply the second fraction by (1/sin^2x) / (1/sin^2x):
1/sin^2x - (cos^2x / sin^2x * 1/sin^2x) = 1
Simplifying further:
1/sin^2x - cos^2x / sin^4x = 1
Now, we can subtract the fractions:
(1 - cos^2x) / sin^2x = 1
Using the identity sin^2x + cos^2x = 1, we can substitute (1 - cos^2x) with sin^2x:
sin^2x / sin^2x = 1
Finally, we have:
1 = 1
Step 2: Solve for x
Since the equation simplifies to 1 = 1, it means that x can take any value, and there is no specific solution for this equation.
Therefore, the original equation csc^2x - cos^2x csc^2x = 1 has infinitely many solutions.
(1/sin^2)(1 -cos^2) = 1
but 1 - cos^2 = sin^2
(1/sin^2)(sin^2) = 1 yes