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# Why are all these false?

1) arcsin (x) = csc (x)

2) ln(x) - ln (y) = ln(x)/ln(y)

3) cos(2x)/2 = cos (x)

### 1 Answer

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- Uncle MichaelLv 74 months ago
1)

arcsin(x) is an angle, θ, where sin(θ) = x

csc(x) = 1/sin(x)

For example, arcsin(1) = π ≈ 3.14,

but csc(1) = 1/sin(1) ≈ 0.84

Hence, arcsin(x) ≠ csc(x)

====

2)

ln(x) - ln(y) = ln(x/y) ≠ ln(x)/ln(y)

For example, when x = 3 and y = 2:

ln(x) - ln(y) = ln(3) - ln(2) ≈ 0.405,

but ln(x)/ln(y) = ln(3)/ln(2) ≈ 1.58

Hence, ln(x/y) ≠ ln(x)/ln(y)

====

3)

cos(2x)/2 = [cos²(x) - sin²(x)]/2 ≠ cos(x)

For example, when x = 30°:

cos(2x)/2 = cos(60°)/2 = 1/4,

but cos(x) = cos(30°) = √3/2

Hence, cos(2x)/2 ≠ cos(x)

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