Question
Download Solution PDFWhat is cot A + cosec A equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
cos 2A = cos2 A - sin2 A = 2 cos2 A - 1
sin 2A = 2 sin A × cos A
Calculation:
\(\Rightarrow \cot A+cosec~A=\frac{\cos A}{\sin A}+\frac{1}{\sin A}\)
\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}\)
As we know that,
cos 2A = cos2 A - sin2 A = 2 cos2 A - 1
\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{\sin A}\)
As we know that,
sin 2A = 2 sin A × cos A
\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}\)
\(=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{\sin A}=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{2\times \cos \left( \frac{A}{2} \right)\times \sin \left( \frac{A}{2} \right)}=\cot \left( \frac{A}{2} \right)\)
∴ The value of cot A + cosec A is \(\cot \left( \frac{A}{2} \right)\)
Last updated on May 30, 2025
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