What is the general solution for arccos(x)+arccos(2x)=arccos(1/2) ?

The general solution for arccos(x)+arccos(2x)=arccos(1/2) is x= 1/2

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arccos(x)+arccos(2x)=arccos(1/2)

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

x = \frac{1}{2}

Solution steps

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

a = b \Rightarrow cos (a) = cos (b)

cos (arccos (x) + arccos (2 x)) = cos (arccos (\frac{1}{2}))

Use the following identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

cos (arccos (x)) cos (arccos (2 x)) - sin (arccos (x)) sin (arccos (2 x)) = cos (arccos (\frac{1}{2}))

Use the following identity:

cos (arccos (x)) = x

Use the following identity:

cos (arccos (x)) = x

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

x \cdot 2 x - 1 - x^{2} 1 - (2 x)^{2} = \frac{1}{2}

Solve

x 2 x - 1 - x^{2} 1 - (2 x)^{2} = \frac{1}{2} : x = \frac{1}{2}, x = - \frac{1}{2}

x = \frac{1}{2}, x = - \frac{1}{2}

Verify solutions by plugging them into the original equation

x = \frac{1}{2}

tan (θ) = \frac{2 3}{3} sin (θ), 0 \leq θ \leq 2 π

x, cos (x + y) = - cos (x)

4 + 2 tan^{2} (x) = 5 tan^{2} (x) + 3

3 sin (x) + cos (x) = 4

θ, sec (θ) = 2

What is the general solution for arccos(x)+arccos(2x)=arccos(1/2) ?
The general solution for arccos(x)+arccos(2x)=arccos(1/2) is x= 1/2