Question

I need the whole solution

Answer 1

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 $\frac{1}{cos}$

$\frac{1}{cos}$ = 1 $\frac{1}{cos}$ = -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 $\frac{1}{sin}$

$\frac{1}{sin}$ = 1 $\frac{1}{sin}$ = -1

cross multiply to get:

sinΘ = 1 sinΘ = -1

use the unit circle (or a calculator) to find that Θ = $\frac{\pi}{2}$ and $\frac{3\pi}{2}$

Answer: 0, π, $\frac{\pi}{2}$ , $\frac{3\pi}{2}$