Discover new perspectives and gain insights with IDNLearn.com's diverse answers. Find reliable solutions to your questions quickly and easily with help from our experienced experts.
Sagot :
Certainly! Let's walk through the verification of the trigonometric identity:
[tex]\[\cot ^2\left(\frac{x}{2}\right)=\frac{\cos x+1}{\cos x-1}\][/tex]
### Step-by-Step Solution:
1. Given:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) \][/tex]
This is our starting expression.
2. Transform Cotangent to Tangent Form:
[tex]\[ \cot\left(\frac{x}{2}\right) = \frac{1}{\tan\left(\frac{x}{2}\right)} \][/tex]
Hence,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \left(\frac{1}{\tan\left(\frac{x}{2}\right)}\right)^2 \][/tex]
3. Expressing [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex]:
By using the half-angle formula for tangent,
[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \][/tex]
So,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \left(\frac{1}{\sqrt{\frac{1 - \cos x}{1 + \cos x}}}\right)^2 \][/tex]
4. Simplify the Square Root Term:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1}{\left(\sqrt{\frac{1 - \cos x}{1 + \cos x}}\right)^2} \][/tex]
Since [tex]\(\left(\sqrt{\frac{1 - \cos x}{1 + \cos x}}\right)^2\)[/tex] simplifies to [tex]\(\frac{1 - \cos x}{1 + \cos x}\)[/tex],
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1}{\frac{1 - \cos x}{1 + \cos x}} \][/tex]
5. Utilize the Multiplicative Inverse (Reciprocal):
By taking the reciprocal,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1 + \cos x}{1 - \cos x} \][/tex]
Thus, we have successfully shown that:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{\cos x + 1}{1 - \cos x} \][/tex]
This completes the proof of the identity.
[tex]\[\cot ^2\left(\frac{x}{2}\right)=\frac{\cos x+1}{\cos x-1}\][/tex]
### Step-by-Step Solution:
1. Given:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) \][/tex]
This is our starting expression.
2. Transform Cotangent to Tangent Form:
[tex]\[ \cot\left(\frac{x}{2}\right) = \frac{1}{\tan\left(\frac{x}{2}\right)} \][/tex]
Hence,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \left(\frac{1}{\tan\left(\frac{x}{2}\right)}\right)^2 \][/tex]
3. Expressing [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex]:
By using the half-angle formula for tangent,
[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \][/tex]
So,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \left(\frac{1}{\sqrt{\frac{1 - \cos x}{1 + \cos x}}}\right)^2 \][/tex]
4. Simplify the Square Root Term:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1}{\left(\sqrt{\frac{1 - \cos x}{1 + \cos x}}\right)^2} \][/tex]
Since [tex]\(\left(\sqrt{\frac{1 - \cos x}{1 + \cos x}}\right)^2\)[/tex] simplifies to [tex]\(\frac{1 - \cos x}{1 + \cos x}\)[/tex],
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1}{\frac{1 - \cos x}{1 + \cos x}} \][/tex]
5. Utilize the Multiplicative Inverse (Reciprocal):
By taking the reciprocal,
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{1 + \cos x}{1 - \cos x} \][/tex]
Thus, we have successfully shown that:
[tex]\[ \cot ^2\left(\frac{x}{2}\right) = \frac{\cos x + 1}{1 - \cos x} \][/tex]
This completes the proof of the identity.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.