Answer :
To determine which given options are true given the equation [tex]\( 2 \tan^2 x - \sec x = 1 \)[/tex], we need to analyze each option in the context of the equation. We'll evaluate each condition one by one.
### Option 1: [tex]\( \sec x = -1 \)[/tex]
1. Substitute [tex]\( \sec x = -1 \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - (-1) = 1 \][/tex]
2. Simplify the equation:
[tex]\[ 2 \tan^2 x + 1 = 1 \][/tex]
3. Subtract 1 from both sides:
[tex]\[ 2 \tan^2 x = 0 \][/tex]
4. Divide both sides by 2:
[tex]\[ \tan^2 x = 0 \][/tex]
5. Take the square root of both sides:
[tex]\[ \tan x = 0 \][/tex]
We now verify whether [tex]\( \sec x = -1 \)[/tex] when [tex]\( \tan x = 0 \)[/tex]. Recall that [tex]\(\sec x = \frac{1}{\cos x}\)[/tex]. If [tex]\(\tan x = 0\)[/tex], then [tex]\( \sin x = 0 \)[/tex] and [tex]\(\cos x = 1\)[/tex]. Thus:
[tex]\[ \sec x = \frac{1}{1} = 1 \][/tex]
This contradicts [tex]\( \sec x = -1 \)[/tex]. Hence, [tex]\(\sec x = -1 \)[/tex] is false.
### Option 2: [tex]\( \sec x = 3 \)[/tex]
1. Substitute [tex]\( \sec x = 3 \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - 3 = 1 \][/tex]
2. Simplify the equation:
[tex]\[ 2 \tan^2 x = 4 \][/tex]
3. Divide both sides by 2:
[tex]\[ \tan^2 x = 2 \][/tex]
4. Take the square root of both sides:
[tex]\[ \tan x = \pm \sqrt{2} \][/tex]
We verify whether [tex]\(\sec x = 3\)[/tex] when [tex]\(\tan x = \pm \sqrt{2}\)[/tex]. Recall that:
[tex]\[ \sec^2 x = 1 + \tan^2 x \][/tex]
For [tex]\(\tan x = \sqrt{2}\)[/tex]:
[tex]\[ \sec^2 x = 1 + 2 = 3 \implies \sec x = \sqrt{3} \neq 3 \][/tex]
Therefore, [tex]\(\sec x = 3 \)[/tex] is false.
### Option 3: [tex]\( \tan x = 3 \)[/tex]
1. Substitute [tex]\( \tan x = 3 \)[/tex] into the equation:
[tex]\[ 2 (3)^2 - \sec x = 1 \][/tex]
2. Simplify the term:
[tex]\[ 2 (9) - \sec x = 1 \][/tex]
[tex]\[ 18 - \sec x = 1 \][/tex]
3. Subtract 18 from both sides:
[tex]\[ - \sec x = -17 \][/tex]
4. Multiply by -1:
[tex]\[ \sec x = 17 \][/tex]
However, this [tex]\(\sec x = 17\)[/tex] does not match any of the provided answers in the options, so [tex]\(\tan x = 3\)[/tex] is false.
### Option 4: [tex]\( \tan x = -1 \)[/tex]
1. Substitute [tex]\( \tan x = -1 \)[/tex] into the equation:
[tex]\[ 2 (-1)^2 - \sec x = 1 \][/tex]
2. Simplify the term:
[tex]\[ 2 (1) - \sec x = 1 \][/tex]
[tex]\[ 2 - \sec x = 1 \][/tex]
3. Subtract 2 from both sides:
[tex]\[ - \sec x = -1 \][/tex]
4. Multiply by -1:
[tex]\[ \sec x = 1 \][/tex]
Since the result does not match any of the provided values in the options, then [tex]\( \tan x = -1 \)[/tex] is false.
### Option 5: [tex]\( \sec x = \frac{3}{2} \)[/tex]
1. Substitute [tex]\( \sec x = \frac{3}{2} \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - \frac{3}{2} = 1 \][/tex]
2. Move [tex]\(\frac{3}{2}\)[/tex] to the right side:
[tex]\[ 2 \tan^2 x = 1 + \frac{3}{2} \][/tex]
3. Simplify:
[tex]\[ 2 \tan^2 x = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \][/tex]
4. Divide by 2:
[tex]\[ \tan^2 x = \frac{5}{4} \][/tex]
[tex]\[ \tan x = \pm \sqrt{\frac{5}{4}} \][/tex]
[tex]\[ \tan x = \pm \frac{\sqrt{5}}{2} \][/tex]
Given this, we verify:
[tex]\[ \sec^2 x = 1 + \tan^2 x \implies \sec^2 x = 1 + \frac{5}{4} = \frac{9}{4} \implies \sec x = \pm \frac{3}{2} \][/tex]
Hence, [tex]\(\sec x = \frac{3}{2}\)[/tex] is true.
### Conclusion:
The correct answer is:
[tex]\[ \sec x = \frac{3}{2} \][/tex]
### Option 1: [tex]\( \sec x = -1 \)[/tex]
1. Substitute [tex]\( \sec x = -1 \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - (-1) = 1 \][/tex]
2. Simplify the equation:
[tex]\[ 2 \tan^2 x + 1 = 1 \][/tex]
3. Subtract 1 from both sides:
[tex]\[ 2 \tan^2 x = 0 \][/tex]
4. Divide both sides by 2:
[tex]\[ \tan^2 x = 0 \][/tex]
5. Take the square root of both sides:
[tex]\[ \tan x = 0 \][/tex]
We now verify whether [tex]\( \sec x = -1 \)[/tex] when [tex]\( \tan x = 0 \)[/tex]. Recall that [tex]\(\sec x = \frac{1}{\cos x}\)[/tex]. If [tex]\(\tan x = 0\)[/tex], then [tex]\( \sin x = 0 \)[/tex] and [tex]\(\cos x = 1\)[/tex]. Thus:
[tex]\[ \sec x = \frac{1}{1} = 1 \][/tex]
This contradicts [tex]\( \sec x = -1 \)[/tex]. Hence, [tex]\(\sec x = -1 \)[/tex] is false.
### Option 2: [tex]\( \sec x = 3 \)[/tex]
1. Substitute [tex]\( \sec x = 3 \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - 3 = 1 \][/tex]
2. Simplify the equation:
[tex]\[ 2 \tan^2 x = 4 \][/tex]
3. Divide both sides by 2:
[tex]\[ \tan^2 x = 2 \][/tex]
4. Take the square root of both sides:
[tex]\[ \tan x = \pm \sqrt{2} \][/tex]
We verify whether [tex]\(\sec x = 3\)[/tex] when [tex]\(\tan x = \pm \sqrt{2}\)[/tex]. Recall that:
[tex]\[ \sec^2 x = 1 + \tan^2 x \][/tex]
For [tex]\(\tan x = \sqrt{2}\)[/tex]:
[tex]\[ \sec^2 x = 1 + 2 = 3 \implies \sec x = \sqrt{3} \neq 3 \][/tex]
Therefore, [tex]\(\sec x = 3 \)[/tex] is false.
### Option 3: [tex]\( \tan x = 3 \)[/tex]
1. Substitute [tex]\( \tan x = 3 \)[/tex] into the equation:
[tex]\[ 2 (3)^2 - \sec x = 1 \][/tex]
2. Simplify the term:
[tex]\[ 2 (9) - \sec x = 1 \][/tex]
[tex]\[ 18 - \sec x = 1 \][/tex]
3. Subtract 18 from both sides:
[tex]\[ - \sec x = -17 \][/tex]
4. Multiply by -1:
[tex]\[ \sec x = 17 \][/tex]
However, this [tex]\(\sec x = 17\)[/tex] does not match any of the provided answers in the options, so [tex]\(\tan x = 3\)[/tex] is false.
### Option 4: [tex]\( \tan x = -1 \)[/tex]
1. Substitute [tex]\( \tan x = -1 \)[/tex] into the equation:
[tex]\[ 2 (-1)^2 - \sec x = 1 \][/tex]
2. Simplify the term:
[tex]\[ 2 (1) - \sec x = 1 \][/tex]
[tex]\[ 2 - \sec x = 1 \][/tex]
3. Subtract 2 from both sides:
[tex]\[ - \sec x = -1 \][/tex]
4. Multiply by -1:
[tex]\[ \sec x = 1 \][/tex]
Since the result does not match any of the provided values in the options, then [tex]\( \tan x = -1 \)[/tex] is false.
### Option 5: [tex]\( \sec x = \frac{3}{2} \)[/tex]
1. Substitute [tex]\( \sec x = \frac{3}{2} \)[/tex] into the equation:
[tex]\[ 2 \tan^2 x - \frac{3}{2} = 1 \][/tex]
2. Move [tex]\(\frac{3}{2}\)[/tex] to the right side:
[tex]\[ 2 \tan^2 x = 1 + \frac{3}{2} \][/tex]
3. Simplify:
[tex]\[ 2 \tan^2 x = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \][/tex]
4. Divide by 2:
[tex]\[ \tan^2 x = \frac{5}{4} \][/tex]
[tex]\[ \tan x = \pm \sqrt{\frac{5}{4}} \][/tex]
[tex]\[ \tan x = \pm \frac{\sqrt{5}}{2} \][/tex]
Given this, we verify:
[tex]\[ \sec^2 x = 1 + \tan^2 x \implies \sec^2 x = 1 + \frac{5}{4} = \frac{9}{4} \implies \sec x = \pm \frac{3}{2} \][/tex]
Hence, [tex]\(\sec x = \frac{3}{2}\)[/tex] is true.
### Conclusion:
The correct answer is:
[tex]\[ \sec x = \frac{3}{2} \][/tex]
