Question

If 2 tangent squared x minus secant x equals 1, which of the following are true? Select all that apply.
(1 point)
Responses

secant x equals negative 1
Image with alt text: secant x equals negative 1

sec x = 3
sec x = 3

tan x = 3
tan x = 3

tangent x equals negative 1
Image with alt text: tangent x equals negative 1

secant x equals start fraction 3 over 2 end fraction

Answer 1

To solve the equation given:

\[ 2 \tan^2 x - \sec x = 1 \]

We'll use the identity:

\[ \sec^2 x = 1 + \tan^2 x \]

From the identity, we can express \(\tan^2 x\) in terms of \(\sec x\):

\[ \tan^2 x = \sec^2 x - 1 \]

Substituting this into the original equation:

\[ 2(\sec^2 x - 1) - \sec x = 1 \]

Expanding this:

\[ 2\sec^2 x - 2 - \sec x = 1 \]

Then we can rearrange the equation:

\[ 2\sec^2 x - \sec x - 3 = 0 \]

This is a quadratic equation in terms of \(\sec x\). Let \(y = \sec x\). The equation becomes:

\[ 2y^2 - y - 3 = 0 \]

To solve for \(y\) using the quadratic formula:

\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a = 2\), \(b = -1\), and \(c = -3\):

\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-3)}}{2(2)} \]
\[ y = \frac{1 \pm \sqrt{1 + 24}}{4} \]
\[ y = \frac{1 \pm \sqrt{25}}{4} \]
\[ y = \frac{1 \pm 5}{4} \]

This gives us two potential solutions:

1. \(y = \frac{6}{4} = \frac{3}{2}\)
2. \(y = \frac{-4}{4} = -1\)

Thus, the possible values for \(\sec x\) are:

1. \(\sec x = \frac{3}{2}\)
2. \(\sec x = -1\)

Next, let's find the corresponding \(\tan x\):

Using \(\sec^2 x = 1 + \tan^2 x\):

1. For \(\sec x = \frac{3}{2}\):

\[ \sec^2 x = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
\[ \frac{9}{4} = 1 + \tan^2 x \]
\[ \tan^2 x = \frac{9}{4} - 1 = \frac{5}{4} \]
\[ \tan x = \pm \frac{\sqrt{5}}{2} \]

2. For \(\sec x = -1\):

\[ \sec^2 x = (-1)^2 = 1 \]
\[ 1 = 1 + \tan^2 x \]
\[ \tan^2 x = 0 \]
\[ \tan x = 0 \]

Now, we can analyze the given options:

1. **secant x equals negative 1** - True
2. **sec x = 3** - False (the solutions for sec are \(\frac{3}{2}\) and \(-1\))
3. **tan x = 3** - False (the possible values for \(\tan x\) are \(\frac{\sqrt{5}}{2}\) or \(0\))
4. **tangent x equals negative 1** - False (none of the values from our findings match \(-1\))
5. **secant x equals \(\frac{3}{2}\)** - True

Thus, the correct answers are:

- **secant x equals negative 1**
- **secant x equals \(\frac{3}{2}\)**