I have to factor the expression and use fundamental identities:
sins^2xcscx^2-sin^2x
sec^4x-tan^4x
sins^2xcscx^2-sin^2x
= sin^2x(csc^2x - 1)
= sin^2x(cot^2x)
= sin^2x(cos^2x/sin^2x)
= cos^2x
sec^4x - tan^4x
= (sec^2x - tan^2x)(sec^2x + tan^2x)
= (tan^2x + 1 - tan^2x)(tan^2x + 1 + tan^2x)
= 1(2tan^2x +1)
= 2tan^2x + 1
OHHH okay, THANKS!
To factor the given expressions and use fundamental identities, let's break down each expression one by one:
1. Factor the expression sins^2xcscx^2-sin^2x:
To factor this expression, we can start by factoring out sin^2x as a common factor:
sin^2x (cscx^2 - 1)
Next, we can use the fundamental identity cscx^2 = 1/sin^2x:
sin^2x (1/sin^2x - 1)
Now, simplify by finding a common denominator for the fractions:
sin^2x ((1 - sin^2x)/sin^2x)
Using the identity 1 - sin^2x = cos^2x, we can substitute it in the expression:
sin^2x (cos^2x/sin^2x)
Finally, cancel out the common factor sin^2x, and the expression simplifies to:
cos^2x
Therefore, the factored expression is cos^2x.
2. Factor the expression sec^4x - tan^4x:
To factor this expression, we can use the identity sec^2x = 1 + tan^2x:
Rewrite the expression using the identity:
(sec^2x)^2 - tan^4x
Substitute sec^2x with 1 + tan^2x in the expression:
(1 + tan^2x)^2 - tan^4x
Expand the square of the binomial:
(1 + 2tan^2x + tan^4x) - tan^4x
Combine like terms and simplify:
1 + 2tan^2x
Therefore, the factored expression is 1 + 2tan^2x.