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Q.
- Apply Double Angle Identity: To verify the identity, we will start by using the double angle identity for cotangent, which is . We will then express and in terms of using the Pythagorean identities.
- Express in Terms of : The double angle formulas for sine and cosine are and . We will use these to express and in terms of .
- Use Pythagorean Identities: Since , we can express as and as . Therefore, and $\cos(\(2\)\theta) = \cos^\(2\)(\theta) - \sin^\(2\)(\theta) = \cos^\(2\)(\theta) - (\tan(\theta)\cos(\theta))^\(2\).
- Rewrite in Terms of \(\tan(\theta)\): We can rewrite \(\cos^2(\theta)\) as \(\frac{1}{1 + \tan^2(\theta)}\) using the Pythagorean identity \(1 + \tan^2(\theta) = \frac{1}{\cos^2(\theta)}\). This gives us \(\sin(2\theta) = \frac{2\tan(\theta)}{1 + \tan^2(\theta)}\) and \(\cos(2\theta) = \frac{1}{1 + \tan^2(\theta)} - \frac{\tan^2(\theta)}{1 + \tan^2(\theta)}\).
- Express \(\cot(2\theta)\): Now we can express \(\cot(2\theta)\) as \(\frac{\cos(2\theta)}{\sin(2\theta)} = \frac{\frac{1}{1 + \tan^2(\theta)} - \frac{\tan^2(\theta)}{1 + \tan^2(\theta)}}{\frac{2\tan(\theta)}{1 + \tan^2(\theta)}}\).
- Simplify the Expression: Simplify the expression by combining the terms in the numerator, which gives us \((1 - \tan^2(\theta))/(1 + \tan^2(\theta))\). Then divide by \((2\tan(\theta)/(1 + \tan^2(\theta)))\) to get \((1 - \tan^2(\theta))/(2\tan(\theta))\).
- Verify the Identity: We have now shown that \(\cot(2\theta) = \frac{1 - \tan^2(\theta)}{2\tan(\theta)}\), which is the same as the given identity. Therefore, the identity is verified.
