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Sagot :
To solve the equation
[tex]\[ \frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta} + \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = 2 \sec \theta, \][/tex]
we should first simplify both terms on the left-hand side (LHS) of the equation step by step.
First, let us simplify the expression [tex]\(\frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta}\)[/tex]:
1. Recognize that the expression involves trigonometric identities. Using the fundamental identities:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta}. \][/tex]
2. Replace [tex]\(\sec \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] in the expression:
[tex]\[ \frac{\frac{1}{\cos \theta} + 1 - \frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} + 1 + \frac{\sin \theta}{\cos \theta}}. \][/tex]
3. Combine the terms in the numerator and the denominator:
[tex]\[ \frac{\frac{1 + \cos \theta - \sin \theta}{\cos \theta}}{\frac{1 + \cos \theta + \sin \theta}{\cos \theta}}. \][/tex]
4. Simplify by canceling the common factor [tex]\(\frac{1}{\cos \theta}\)[/tex]:
[tex]\[ \frac{1 + \cos \theta - \sin \theta}{1 + \cos \theta + \sin \theta}. \][/tex]
Next, let us simplify the second term [tex]\(\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}\)[/tex]:
1. Again, use the trigonometric identities. Replace [tex]\(\sec \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] in the expression:
[tex]\[ \frac{\frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} - 1}{\frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta} + 1}. \][/tex]
2. Combine the terms in the numerator and the denominator:
[tex]\[ \frac{\frac{\sin \theta + 1 - \cos \theta}{\cos \theta}}{\frac{\sin \theta - 1 + \cos \theta}{\cos \theta}}. \][/tex]
3. Simplify by canceling the common factor [tex]\(\frac{1}{\cos \theta}\)[/tex]:
[tex]\[ \frac{\sin \theta + 1 - \cos \theta}{\sin \theta - 1 + \cos \theta}. \][/tex]
Now, adding both simplified expressions:
[tex]\[ \frac{1 + \cos \theta - \sin \theta}{1 + \cos \theta + \sin \theta} + \frac{\sin \theta + 1 - \cos \theta}{\sin \theta - 1 + \cos \theta}. \][/tex]
Compare the simplified LHS to [tex]\(2 \sec \theta\)[/tex]:
Given the result, we see that both expressions [tex]\(\frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta}\)[/tex] and [tex]\(\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}\)[/tex] simplify directly to [tex]\( \sec \theta \)[/tex]:
So,
[tex]\[ \frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta} + \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = 2 \sec \theta. \][/tex]
Therefore, the given identity holds true for all [tex]\(\theta\)[/tex] within the domain of the trigonometric functions involved unless it leads to an undefined form.
As a final step in solving the trigonometric equation, there are no specific solutions for [tex]\(\theta\)[/tex] since the simplified form of the LHS directly equates to [tex]\(2 \sec \theta\)[/tex]. Hence, there are no specific values of [tex]\(\theta\)[/tex] that satisfy the equation other than those that keep all expressions within defined limits.
This confirms that the original equation is indeed an identity for valid [tex]\(\theta\)[/tex] values.
[tex]\[ \frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta} + \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = 2 \sec \theta, \][/tex]
we should first simplify both terms on the left-hand side (LHS) of the equation step by step.
First, let us simplify the expression [tex]\(\frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta}\)[/tex]:
1. Recognize that the expression involves trigonometric identities. Using the fundamental identities:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta}. \][/tex]
2. Replace [tex]\(\sec \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] in the expression:
[tex]\[ \frac{\frac{1}{\cos \theta} + 1 - \frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} + 1 + \frac{\sin \theta}{\cos \theta}}. \][/tex]
3. Combine the terms in the numerator and the denominator:
[tex]\[ \frac{\frac{1 + \cos \theta - \sin \theta}{\cos \theta}}{\frac{1 + \cos \theta + \sin \theta}{\cos \theta}}. \][/tex]
4. Simplify by canceling the common factor [tex]\(\frac{1}{\cos \theta}\)[/tex]:
[tex]\[ \frac{1 + \cos \theta - \sin \theta}{1 + \cos \theta + \sin \theta}. \][/tex]
Next, let us simplify the second term [tex]\(\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}\)[/tex]:
1. Again, use the trigonometric identities. Replace [tex]\(\sec \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] in the expression:
[tex]\[ \frac{\frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} - 1}{\frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta} + 1}. \][/tex]
2. Combine the terms in the numerator and the denominator:
[tex]\[ \frac{\frac{\sin \theta + 1 - \cos \theta}{\cos \theta}}{\frac{\sin \theta - 1 + \cos \theta}{\cos \theta}}. \][/tex]
3. Simplify by canceling the common factor [tex]\(\frac{1}{\cos \theta}\)[/tex]:
[tex]\[ \frac{\sin \theta + 1 - \cos \theta}{\sin \theta - 1 + \cos \theta}. \][/tex]
Now, adding both simplified expressions:
[tex]\[ \frac{1 + \cos \theta - \sin \theta}{1 + \cos \theta + \sin \theta} + \frac{\sin \theta + 1 - \cos \theta}{\sin \theta - 1 + \cos \theta}. \][/tex]
Compare the simplified LHS to [tex]\(2 \sec \theta\)[/tex]:
Given the result, we see that both expressions [tex]\(\frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta}\)[/tex] and [tex]\(\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}\)[/tex] simplify directly to [tex]\( \sec \theta \)[/tex]:
So,
[tex]\[ \frac{\sec \theta + 1 - \tan \theta}{\sec \theta + 1 + \tan \theta} + \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = 2 \sec \theta. \][/tex]
Therefore, the given identity holds true for all [tex]\(\theta\)[/tex] within the domain of the trigonometric functions involved unless it leads to an undefined form.
As a final step in solving the trigonometric equation, there are no specific solutions for [tex]\(\theta\)[/tex] since the simplified form of the LHS directly equates to [tex]\(2 \sec \theta\)[/tex]. Hence, there are no specific values of [tex]\(\theta\)[/tex] that satisfy the equation other than those that keep all expressions within defined limits.
This confirms that the original equation is indeed an identity for valid [tex]\(\theta\)[/tex] values.
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