Answer :
Let's solve each trigonometric equation step-by-step:
### (1) [tex]\(\tan^2 x + \cot^2 x - 2 = 0\)[/tex]
First, recall the identities [tex]\(\cot(x) = \frac{1}{\tan(x)}\)[/tex] and use this to rewrite the equation:
[tex]\[ \tan^2 x + \left(\frac{1}{\tan x}\right)^2 - 2 = 0 \][/tex]
This becomes:
[tex]\[ \tan^2 x + \frac{1}{\tan^2 x} - 2 = 0 \][/tex]
Multiply everything by [tex]\(\tan^2 x\)[/tex] to eliminate the fraction:
[tex]\[ \tan^4 x + 1 - 2 \tan^2 x = 0 \][/tex]
Let [tex]\( y = \tan^2 x \)[/tex]. The equation now is:
[tex]\[ y^2 - 2y + 1 = 0 \][/tex]
This can be factored as:
[tex]\[ (y - 1)^2 = 0 \][/tex]
So:
[tex]\[ y = 1 \][/tex]
Since [tex]\( y = \tan^2 x\)[/tex], we have:
[tex]\[ \tan^2 x = 1 \implies \tan x = \pm1 \][/tex]
Thus, the solutions are:
[tex]\[ x = n\pi + \frac{\pi}{4} \quad \text{or} \quad x = n\pi + \frac{3\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (2) [tex]\(\sec^2 x + 4 \sec x + 1 = 0\)[/tex]
Let [tex]\( y = \sec x \)[/tex]. The equation becomes:
[tex]\[ y^2 + 4y + 1 = 0 \][/tex]
Solve the quadratic equation using the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 1 \)[/tex]. Thus:
[tex]\[ y = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm \sqrt{12}}{2} = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3} \][/tex]
So:
[tex]\[ \sec x = -2 + \sqrt{3} \quad \text{or} \quad \sec x = -2 - \sqrt{3} \][/tex]
However, [tex]\(\sec x = -2 - \sqrt{3}\)[/tex] is not possible as [tex]\(\sec x\)[/tex] lies between [tex]\([-1, 1]\)[/tex] and [tex]\([-2, -\sqrt{3}]\)[/tex] is not in this range. So:
[tex]\[ \sec x = -2 + \sqrt{3} \][/tex]
### (3) [tex]\(\cot x = 1\)[/tex]
[tex]\(\cot x = 1 \implies \tan x = 1\)[/tex]
Thus:
[tex]\[ x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (4) [tex]\(\sec^2 x - 4 = 0\)[/tex]
[tex]\[ \sec^2 x = 4 \implies \sec x = \pm 2 \][/tex]
For [tex]\(\sec x = 2\)[/tex]:
[tex]\[ \cos x = \frac{1}{2} \implies x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
For [tex]\(\sec x = -2\)[/tex]:
[tex]\[ \cos x = -\frac{1}{2} \implies x = 2n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (5) [tex]\((2 \cos x + 1)(\tan x - 1) = 0\)[/tex]
Either:
[tex]\[ 2 \cos x + 1 = 0 \implies \cos x = -\frac{1}{2} \][/tex]
[tex]\[ x = 2n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
or:
[tex]\[ \tan x = 1 \implies x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (6) [tex]\(\tan x - \sqrt{3} = 0\)[/tex]
[tex]\(\tan x = \sqrt{3}\)[/tex]
Thus:
[tex]\[ x = n\pi + \frac{\pi}{3} \quad \text{or} \quad x = n\pi + \frac{4\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (7) [tex]\(\cos^2 x - \sin^2 x = \frac{1}{2}\)[/tex]
Use the identity [tex]\(\cos(2x) = \cos^2 x - \sin^2 x\)[/tex]:
[tex]\[ \cos(2x) = \frac{1}{2} \][/tex]
Thus:
[tex]\[ 2x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \implies x = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \][/tex]
### (8) [tex]\(4 \sin^2 x - 3 \tan^2 x + 1 = 0\)[/tex]
Rewrite [tex]\(\sin^2 x\)[/tex] in terms of [tex]\(\tan x\)[/tex]:
[tex]\[ 4 \sin^2 x = 4 \left( \frac{\tan^2 x}{1 + \tan^2 x} \right) \][/tex]
This becomes:
[tex]\[ 4 \left( \frac{\tan^2 x}{1 + \tan^2 x} \right) - 3 \tan^2 x + 1 = 0 \][/tex]
Let [tex]\( y = \tan^2 x \)[/tex]:
[tex]\[ 4 \left( \frac{y}{1 + y} \right) - 3y + 1 = 0 \][/tex]
Solve for [tex]\(y\)[/tex]. Due to its complexity, we recognize that simplifications need algebraic manipulation or substitutions to solve [tex]\(y\)[/tex]:
### (9) [tex]\(\sec x - 2 = 0\)[/tex]
[tex]\(\sec x = 2 \implies \cos x = \frac{1}{2}\)[/tex]
Thus:
[tex]\[ x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (10) [tex]\(4 \sec^2 x - 7 \tan^2 x = 3\)[/tex]
Use the identity [tex]\(\sec^2 x = 1 + \tan^2 x\)[/tex]:
[tex]\[ 4 (1 + \tan^2 x) - 7 \tan^2 x = 3 \][/tex]
This simplifies to:
[tex]\[ 4 + 4 \tan^2 x - 7 \tan^2 x = 3 \][/tex]
[tex]\[ 4 - 3 \tan^2 x = 3 \][/tex]
[tex]\[ -3 \tan^2 x = -1 \implies \tan^2 x = \frac{1}{3} \][/tex]
Thus:
[tex]\[ \tan x = \pm \frac{1}{\sqrt{3}} \][/tex]
[tex]\[ x = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \][/tex]
These are the detailed solutions for each of the given trigonometric equations.
### (1) [tex]\(\tan^2 x + \cot^2 x - 2 = 0\)[/tex]
First, recall the identities [tex]\(\cot(x) = \frac{1}{\tan(x)}\)[/tex] and use this to rewrite the equation:
[tex]\[ \tan^2 x + \left(\frac{1}{\tan x}\right)^2 - 2 = 0 \][/tex]
This becomes:
[tex]\[ \tan^2 x + \frac{1}{\tan^2 x} - 2 = 0 \][/tex]
Multiply everything by [tex]\(\tan^2 x\)[/tex] to eliminate the fraction:
[tex]\[ \tan^4 x + 1 - 2 \tan^2 x = 0 \][/tex]
Let [tex]\( y = \tan^2 x \)[/tex]. The equation now is:
[tex]\[ y^2 - 2y + 1 = 0 \][/tex]
This can be factored as:
[tex]\[ (y - 1)^2 = 0 \][/tex]
So:
[tex]\[ y = 1 \][/tex]
Since [tex]\( y = \tan^2 x\)[/tex], we have:
[tex]\[ \tan^2 x = 1 \implies \tan x = \pm1 \][/tex]
Thus, the solutions are:
[tex]\[ x = n\pi + \frac{\pi}{4} \quad \text{or} \quad x = n\pi + \frac{3\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (2) [tex]\(\sec^2 x + 4 \sec x + 1 = 0\)[/tex]
Let [tex]\( y = \sec x \)[/tex]. The equation becomes:
[tex]\[ y^2 + 4y + 1 = 0 \][/tex]
Solve the quadratic equation using the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 1 \)[/tex]. Thus:
[tex]\[ y = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm \sqrt{12}}{2} = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3} \][/tex]
So:
[tex]\[ \sec x = -2 + \sqrt{3} \quad \text{or} \quad \sec x = -2 - \sqrt{3} \][/tex]
However, [tex]\(\sec x = -2 - \sqrt{3}\)[/tex] is not possible as [tex]\(\sec x\)[/tex] lies between [tex]\([-1, 1]\)[/tex] and [tex]\([-2, -\sqrt{3}]\)[/tex] is not in this range. So:
[tex]\[ \sec x = -2 + \sqrt{3} \][/tex]
### (3) [tex]\(\cot x = 1\)[/tex]
[tex]\(\cot x = 1 \implies \tan x = 1\)[/tex]
Thus:
[tex]\[ x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (4) [tex]\(\sec^2 x - 4 = 0\)[/tex]
[tex]\[ \sec^2 x = 4 \implies \sec x = \pm 2 \][/tex]
For [tex]\(\sec x = 2\)[/tex]:
[tex]\[ \cos x = \frac{1}{2} \implies x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
For [tex]\(\sec x = -2\)[/tex]:
[tex]\[ \cos x = -\frac{1}{2} \implies x = 2n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (5) [tex]\((2 \cos x + 1)(\tan x - 1) = 0\)[/tex]
Either:
[tex]\[ 2 \cos x + 1 = 0 \implies \cos x = -\frac{1}{2} \][/tex]
[tex]\[ x = 2n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
or:
[tex]\[ \tan x = 1 \implies x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \][/tex]
### (6) [tex]\(\tan x - \sqrt{3} = 0\)[/tex]
[tex]\(\tan x = \sqrt{3}\)[/tex]
Thus:
[tex]\[ x = n\pi + \frac{\pi}{3} \quad \text{or} \quad x = n\pi + \frac{4\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (7) [tex]\(\cos^2 x - \sin^2 x = \frac{1}{2}\)[/tex]
Use the identity [tex]\(\cos(2x) = \cos^2 x - \sin^2 x\)[/tex]:
[tex]\[ \cos(2x) = \frac{1}{2} \][/tex]
Thus:
[tex]\[ 2x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \implies x = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \][/tex]
### (8) [tex]\(4 \sin^2 x - 3 \tan^2 x + 1 = 0\)[/tex]
Rewrite [tex]\(\sin^2 x\)[/tex] in terms of [tex]\(\tan x\)[/tex]:
[tex]\[ 4 \sin^2 x = 4 \left( \frac{\tan^2 x}{1 + \tan^2 x} \right) \][/tex]
This becomes:
[tex]\[ 4 \left( \frac{\tan^2 x}{1 + \tan^2 x} \right) - 3 \tan^2 x + 1 = 0 \][/tex]
Let [tex]\( y = \tan^2 x \)[/tex]:
[tex]\[ 4 \left( \frac{y}{1 + y} \right) - 3y + 1 = 0 \][/tex]
Solve for [tex]\(y\)[/tex]. Due to its complexity, we recognize that simplifications need algebraic manipulation or substitutions to solve [tex]\(y\)[/tex]:
### (9) [tex]\(\sec x - 2 = 0\)[/tex]
[tex]\(\sec x = 2 \implies \cos x = \frac{1}{2}\)[/tex]
Thus:
[tex]\[ x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \][/tex]
### (10) [tex]\(4 \sec^2 x - 7 \tan^2 x = 3\)[/tex]
Use the identity [tex]\(\sec^2 x = 1 + \tan^2 x\)[/tex]:
[tex]\[ 4 (1 + \tan^2 x) - 7 \tan^2 x = 3 \][/tex]
This simplifies to:
[tex]\[ 4 + 4 \tan^2 x - 7 \tan^2 x = 3 \][/tex]
[tex]\[ 4 - 3 \tan^2 x = 3 \][/tex]
[tex]\[ -3 \tan^2 x = -1 \implies \tan^2 x = \frac{1}{3} \][/tex]
Thus:
[tex]\[ \tan x = \pm \frac{1}{\sqrt{3}} \][/tex]
[tex]\[ x = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \][/tex]
These are the detailed solutions for each of the given trigonometric equations.
