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Question

# Solve each equation for x, where x in [0, 2pi].a) sec x csc x - 2 csc x = 0b) 3 sec ^2 x - 4 = 0c) 2 sin x sec x - 2 square root of 3 sin x = 0d) 2 cot x + sec^2 x = 0e) cot x csc^2 x = 2 cot xf) 3 tan^3 x - tan x = 0

Solution

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(a) We would like to solve the equation $\color{#4257b2}\sec x\csc x-2\csc x=0$ for $\color{#4257b2}0 \leq x \leq 2\pi$. First, we note that all terms contain $\color{#4257b2}\csc x$, so we can take it as a common factor.

$\sec x\csc x-2\csc x=0$

$\csc x\left(\sec x-2\right)=0$

Now we can use the zero-factor property to find the values of $\color{#4257b2}x$.

$\csc x=0\ \ \ \text{or}\ \ \ \sec x-2=0$

$\csc x=0\ \ \ \text{or}\ \ \ \sec x=2$

But we know that $\color{#4257b2}|\csc x| \geq 1$, so the solution $\color{#4257b2}\csc x=0$ is refused.

$\sec x=2$

But we know that $\color{#4257b2}\sec x=\dfrac{1}{\cos x}$, so we can use this identity in our equation.

$\dfrac{1}{\cos x}=2$

$\cos x=\dfrac{1}{2}$

$\cos^{-1}\left(\cos x\right)=\cos^{-1}\left(\dfrac{1}{2}\right)$

$x=\cos^{-1}\left(\dfrac{1}{2}\right)$

$x=\dfrac{\pi}{3}$

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