sin^4t-cos^4t/sin^2t cos^2t= sec^2t-csc^2t
i have =(sin^2t+cos^2t)(sin^2t+cos^2t)/sin^2tcos^2t
then do i go =(sin^2t+cos^2t)/sin^2tcos^2t
stumped
LS = (sin^2 t + cos^2 t)(sin^2 t - cos^2 t)/(sin^2 t cos^2 t)
= 1(sin^2 t - cos^2 t)/(sin^2 t cos^2 t)
= sin^2 t/(sin^2 t cos^2 t) - cos^2 t/(sin^2 t cos^2 t)
= 1/cos^2 t - 1/sin^2 t
= sec^2 t - csc^2 t
= RS
To simplify the expression (sin^4t - cos^4t)/(sin^2t * cos^2t), you can start by factoring the numerator as a difference of squares:
(sin^2t - cos^2t)(sin^2t + cos^2t)/(sin^2t * cos^2t)
Next, you can simplify the expression by using trigonometric identities:
(sin^2t - cos^2t) = -cos^2t + sin^2t = -cos^2t + (1 - cos^2t) = 1 - 2cos^2t
Replacing this in the expression, we have:
(1 - 2cos^2t)(sin^2t + cos^2t)/(sin^2t * cos^2t)
Now, cancel out the common terms:
1 - 2cos^2t)/(sin^2t * cos^2t)
To further simplify, you can use reciprocal identities:
1/sin^2t = csc^2t and 1/cos^2t = sec^2t, so we have:
(sec^2t - 2cos^2t)/csc^2t
Finally, you can simplify further by using the Pythagorean identity sin^2t + cos^2t = 1:
(sec^2t - 2cos^2t)/csc^2t = sec^2t - 2cos^2t = sec^2t - (1 - sin^2t) = sec^2t - 1 + sin^2t = sec^2t - 1 + (1 - cos^2t) = sec^2t - cos^2t = sec^2t - csc^2t
Therefore, (sin^4t - cos^4t)/(sin^2t * cos^2t) simplifies to sec^2t - csc^2t.
To simplify the expression (sin^4t - cos^4t) / (sin^2t * cos^2t), you can follow these steps:
Step 1: Simplify the numerator.
The numerator, (sin^4t - cos^4t), can be factored as a difference of squares:
(sin^2t + cos^2t)(sin^2t - cos^2t).
Step 2: Apply the identity sin^2t + cos^2t = 1.
Replace (sin^2t + cos^2t) in the numerator with 1:
1 * (sin^2t - cos^2t).
Step 3: Simplify the denominator.
The denominator, (sin^2t * cos^2t), can be rewritten as (sin^2t)(cos^2t).
Step 4: Apply the identity sin^2t = 1 - cos^2t.
Replace (sin^2t) in the denominator with (1 - cos^2t):
(1 - cos^2t)(cos^2t).
Step 5: Cancel out common factors.
The (sin^2t - cos^2t) term in the numerator and the (1 - cos^2t) term in the denominator are conjugates, so they cancel each other out.
After canceling out the common factors, you are left with:
1 / cos^2t.
Step 6: Apply the identity sec^2t = 1 / cos^2t.
Since 1 / cos^2t is equal to sec^2t, you can rewrite the expression as:
sec^2t.
Therefore, the simplified expression is sec^2t, which is equivalent to the right side of the equation sec^2t - csc^2t.