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To solve the trigonometric equation [tex]\(2 \cot A = \cot \frac{A}{2} - \tan \frac{A}{2}\)[/tex], we can break down the problem step by step:
1. Expression of cotangent and tangent in terms of sine and cosine:
- The cotangent function in terms of sine and cosine is [tex]\(\cot(x) = \frac{\cos(x)}{\sin(x)}\)[/tex].
- The tangent function in terms of sine and cosine is [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex].
2. Rewrite the given equation using these identities:
- We start with:
[tex]\[ 2 \cot(A) = \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) \][/tex]
- Substituting the trigonometric identities, we have:
[tex]\[ 2 \left(\frac{\cos(A)}{\sin(A)}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} \][/tex]
3. Simplify the right-hand side:
- Simplify the expressions on the right-hand side:
[tex]\[ \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} = \frac{\cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \][/tex]
- Recall that [tex]\(\cos^2(x) - \sin^2(x) = \cos(2x)\)[/tex].
4. Rewrite the equation accordingly:
- Using the cosine double-angle identity, we get:
[tex]\[ 2 \frac{\cos(A)}{\sin(A)} = \frac{\cos(A)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \][/tex]
5. Introduce symmetry and equilibrium:
- Multiply both sides by [tex]\(\sin(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)\)[/tex] to clear fractions:
[tex]\[ 2 \cos(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \cos(A) \sin(A) \][/tex]
6. Compare and further simplify:
- Observe that [tex]\(2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \sin(A)\)[/tex]:
[tex]\[ 2 \cos(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \cos(A) \sin(A) \][/tex]
- Which simplifies to:
[tex]\[ \cos(A) \sin(A) = \cos(A) \sin(A) \][/tex]
At this stage, we see that our simplified equation:
[tex]\[ \cos(A) \sin(A) = \cos(A) \sin(A) \][/tex]
is always true for all [tex]\(\cos(A) \sin(A)\)[/tex].
7. Conclusion:
- This means that the original equation [tex]\(2 \cot A = \cot \frac{A}{2} - \tan \frac{A}{2}\)[/tex] does not have specific solutions for [tex]\(A\)[/tex] that restrict it to a particular set of values.
- Therefore, there are no explicit solutions that restrict [tex]\(A\)[/tex] to specific values beyond the general trigonometric cycles.
So, the equation [tex]\(2 \cot(A) = \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right)\)[/tex] has no specific solution sets.
1. Expression of cotangent and tangent in terms of sine and cosine:
- The cotangent function in terms of sine and cosine is [tex]\(\cot(x) = \frac{\cos(x)}{\sin(x)}\)[/tex].
- The tangent function in terms of sine and cosine is [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex].
2. Rewrite the given equation using these identities:
- We start with:
[tex]\[ 2 \cot(A) = \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) \][/tex]
- Substituting the trigonometric identities, we have:
[tex]\[ 2 \left(\frac{\cos(A)}{\sin(A)}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} \][/tex]
3. Simplify the right-hand side:
- Simplify the expressions on the right-hand side:
[tex]\[ \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} = \frac{\cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \][/tex]
- Recall that [tex]\(\cos^2(x) - \sin^2(x) = \cos(2x)\)[/tex].
4. Rewrite the equation accordingly:
- Using the cosine double-angle identity, we get:
[tex]\[ 2 \frac{\cos(A)}{\sin(A)} = \frac{\cos(A)}{\sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \][/tex]
5. Introduce symmetry and equilibrium:
- Multiply both sides by [tex]\(\sin(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)\)[/tex] to clear fractions:
[tex]\[ 2 \cos(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \cos(A) \sin(A) \][/tex]
6. Compare and further simplify:
- Observe that [tex]\(2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \sin(A)\)[/tex]:
[tex]\[ 2 \cos(A) \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = \cos(A) \sin(A) \][/tex]
- Which simplifies to:
[tex]\[ \cos(A) \sin(A) = \cos(A) \sin(A) \][/tex]
At this stage, we see that our simplified equation:
[tex]\[ \cos(A) \sin(A) = \cos(A) \sin(A) \][/tex]
is always true for all [tex]\(\cos(A) \sin(A)\)[/tex].
7. Conclusion:
- This means that the original equation [tex]\(2 \cot A = \cot \frac{A}{2} - \tan \frac{A}{2}\)[/tex] does not have specific solutions for [tex]\(A\)[/tex] that restrict it to a particular set of values.
- Therefore, there are no explicit solutions that restrict [tex]\(A\)[/tex] to specific values beyond the general trigonometric cycles.
So, the equation [tex]\(2 \cot(A) = \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right)\)[/tex] has no specific solution sets.
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