What is sqrt((2csc(θ))^2-4) ?

The solution to sqrt((2csc(θ))^2-4) is 2cot(θ)

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simplify sqrt((2csc(θ))^2-4)

simplify

(2 csc (θ))^{2} - 4

2 cot (θ)

Solution steps

(2 csc (θ))^{2} - 4

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} csc^{2} (θ) - 4

Factor

2^{2} csc^{2} (θ) - 4 : 4 (csc^{2} (θ) - 1)

= 4 (csc^{2} (θ) - 1)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 4 csc^{2} (θ) - 1

4 = 2

= 2 csc^{2} (θ) - 1

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

csc^{2} (x) - 1 = cot^{2} (x)

= 2 cot^{2} (θ)

cot^{2} (θ) = cot (θ)

= 2 cot (θ)

f (x) = cos (9 0^{\circ} + arcsin (x))

sin (arccos (x + 3))

cos (- x) sin (- x) csc (- x)

4 sin (3 0^{\circ} + x)

7 sec^{2} (x) - 7