Confused what the question is. Are you looking for the product or the zeroes?
If you are looking for the product, then:
Use foil to get: sec²(1) - sec²(-csc²) -1(1) -1(-csc²)
= sec² + sec²csc² - 1 + csc²
= sec²csc² + sec² + csc² - 1
= sec²csc² + 1 - 1 (NOTE: sec² + csc² = 1 is an identity)
= sec²csc²
Answer: sec²csc²
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If you are looking for the zeroes, then:
Using the zero product property, set each factor equal to zero and solve.
First factor:
sec²Θ - 1 = 0
sec²Θ = 1
secΘ = 1, -1
remember that secΘ is [tex] \frac{1}{cos} [/tex]
[tex] \frac{1}{cos} [/tex] = 1 [tex] \frac{1}{cos} [/tex] = -1
cross multiply to get:
cosΘ = 1 cosΘ = -1
use the unit circle (or a calculator) to find that Θ = 0 and π
Second factor:
1 - csc²Θ = 0
1 = csc²Θ
1, -1 = cscΘ
remember that cscΘ is [tex] \frac{1}{sin} [/tex]
[tex] \frac{1}{sin} [/tex] = 1 [tex] \frac{1}{sin} [/tex] = -1
cross multiply to get:
sinΘ = 1 sinΘ = -1
use the unit circle (or a calculator) to find that Θ = [tex] \frac{\pi}{2} [/tex] and [tex] \frac{3\pi}{2} [/tex]
Answer: 0, π, [tex] \frac{\pi}{2} [/tex] , [tex] \frac{3\pi}{2} [/tex]
