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  • <img alt="what\ is\ the\ value\ of\ \cot(2x) \cot(x) \cot(\frac{x}{2}) ?" src="https://learn.careers360.com/latex-image/?what%5C%20is%5C%20the%5C%20value%5C%20of%5C%20%5Ccot%282x%29%20%5Ccot%28x%29

what\ is\ the\ value\ of\ \cot(2x) \cot(x) \cot(\frac{x}{2}) ?

Option: 1

\frac{(\cot^2(x)-1)(1+\cos x)}{\sin x}\\


Option: 2

\frac{(\cot^2(x)-1)(1+\cos x)}{2\sin x}\\


Option: 3

\frac{(\cot^2(x)-1)(1+\cos x)}{4\sin x}\\


Option: 4

\frac{(\cot^2(x)+1)(1+\cos x)}{2\sin x}\\


Answers (1)

best_answer

Trigonometric Ratio for Compound Angles (Part 3) -

Trigonometric Ratio for Compound Angles (Part 3)

 

Proof Cotangent of the Sum and  Difference of two Angles
\\\mathrm{\cot (\alpha+\beta)=\frac{\cot \alpha\cot \beta-1}{\cot \alpha+ \cot \beta}}\\\\\mathrm{\cot (\alpha-\beta)=\frac{\cot \alpha\cot \beta+1}{\cot \alpha- \cot \beta}}

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\cot(2x)= \frac{\cot^2(x)-1}{2cot(x)}... (i)\\ \cot(\frac{x}{2})=\frac{\cos(\frac{x}{2})}{\sin\frac{x}{2}}\\ \ \ \ \ multiply\ and\ devide\ by\ 2\cos(\frac{x}{2})\\ \cot(\frac{x}{2})=\frac{2\cos^2(\frac{x}{2})}{2\sin\frac{x}{2}{\cos(\frac{x}{2})}}\\ \cot(\frac{x}{2})=\frac{1+\cos x}{\sin x} .... (ii)\\ multiply equaition\ (i)\ and\ equation\ (ii)\\ \cot(2x)\cot(\frac{x}{2})=\frac{ \cot^2(x)-1}{2cot(x)} *\frac{1+\cos x}{\sin x}\\ \cot (2x) \cot (\frac{x}{2}) \cot(x) = \frac{(\cot^2(x)-1)(1+\cos x)}{2\sin x}\\

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