Test Index

Trigonometric Ratios and Identities

Section: Mathematics

Question : 4 of 50

Marks: +1, -0

\csc 48^{\circ}+\csc 96^{\circ}+\csc 192^{\circ}+\csc 384^{\circ}=

Solution:

\;\csc 48^{\circ}+\csc 96^{\circ}+\csc 192^{\circ}+\csc 384^{\circ}

\;\because\;\; \csc\theta=\frac{\cot\theta}{2}-\cot\theta

\;\therefore \csc 48^{\circ}=\cot 24^{\circ}-\cot 48^{\circ}

\;\csc 96^{\circ}=\cot 48^{\circ}-\cot 96^{\circ}

\;\csc 192^{\circ}=\cot 96^{\circ}-\cot 192^{\circ}

\;\csc 384^{\circ}=\csc 24^{\circ}

\;=\cot 24^{\circ}-\cot 48^{\circ}+\cot 48^{\circ}-\cot 96^{\circ}+\cot 96^{\circ}-\cot 192^{\circ}+\cos 24^{\circ}

\;=\cot 24^{\circ}-\cot 192^{\circ}+\cos 24^{\circ}

\;\;\text{ and }\; \csc 24^{\circ}=\cot 12^{\circ}-\cot 24^{\circ}

\;=\cot 24^{\circ}-\cot 192^{\circ}+\cot 12^{\circ}-\cot 24^{\circ}

\;=\cot 12^{\circ}-\cot 192^{\circ}

\;=\cot 12^{\circ}-\cot (180+12^{\circ})

\;=\cot 12^{\circ}-\cot 12^{\circ}=0

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