Final answer:
The simplified expression of sec^2(π/2 - x) [sin^2(x) - sin^4(x)] using trigonometric identities is cos^2(x).
Step-by-step explanation:
We need to use trigonometric identities to simplify the expression sec^2(π/2 − x) [sin^2(x) − sin^4(x)].
Firstly, from trigonometric identities we know that:
- sec(θ) = 1/cos(θ)
- cos(π/2 − x) = sin(x)
So, sec^2(π/2 − x) becomes (1/sin(x))^2 or csc^2(x).
Now let's consider the expression [sin^2(x) − sin^4(x)]. This is a difference of squares which can be factored as (sin^2(x))(1 − sin^2(x)).
Remembering another trigonometric identity, where sin^2(x) + cos^2(x) = 1, we can replace (1 − sin^2(x)) with cos^2(x).
Therefore, sin^2(x)(1 − sin^2(x)) simplifies to sin^2(x)cos^2(x).
When we multiply this with csc^2(x), we see that sin^2(x) will cancel out with csc^2(x), since csc(x) = 1/sin(x).
Hence, the simplified expression is just cos^2(x).