We need to solve the equation \(3 \sec^4\theta + 8 = 10 \sec^2\theta\) to find the possible values of \(\tan\theta\).

First, let's introduce a substitution. Let \(x = \sec^2\theta\). The equation becomes:

\(3x^2 + 8 = 10x\)

Rearrange the equation:

\(3x^2 - 10x + 8 = 0\)

This is a quadratic equation. We can solve it using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For our equation, \(a = 3\), \(b = -10\), and \(c = 8\).

First, calculate the discriminant:

\(b^2 - 4ac = (-10)^2 - 4 \times 3 \times 8\)

\(= 100 - 96 = 4\)

Now calculate the roots:

\(x = \frac{10 \pm \sqrt{4}}{6}\)

\(x = \frac{10 \pm 2}{6}\)

This gives us two solutions:

• \(x = \frac{12}{6} = 2\)
• \(x = \frac{8}{6} = \frac{4}{3}\)

Since \(x = \sec^2\theta\), we have \(\sec^2\theta = 2\) or \(\sec^2\theta = \frac{4}{3}\). We know the identity:

\(\tan^2\theta = \sec^2\theta - 1\)

1. If \(\sec^2\theta = 2\), then:

\(\tan^2\theta = 2 - 1 = 1\)

Thus, \(\tan\theta = \pm 1\).

2. If \(\sec^2\theta = \frac{4}{3}\), then:

\(\tan^2\theta = \frac{4}{3} - 1 = \frac{1}{3}\)

Thus, \(\tan\theta = \pm \frac{1}{\sqrt{3}}\).

The values of \(\tan\theta\) can be \(1, \frac{1}{\sqrt{3}}\).

Hence, the correct answer is

$1,\frac{1}{\sqrt{3}}$

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