Question
Prove that:
$\cos15^\circ\cos35^\circ\text{cosec }55^\circ\cos60^\circ\text{cosec }75^\circ=\frac{1}{2}$
$\cos15^\circ\cos35^\circ\text{cosec }55^\circ\cos60^\circ\text{cosec }75^\circ=\frac{1}{2}$
✓
Answer
$\text{L.H.S.}=\cos15^\circ\cos35^\circ\text{cosec }55^\circ\cos60^\circ\text{cosec }75^\circ$
$=\cos(90^\circ-75^\circ)\cos(90^\circ-55^\circ)\frac{1}{\sin55^\circ}\times\frac12\times\frac{1}{\sin75^\circ}$
$=\sin75^\circ\sin55^\circ\frac{1}{\sin55^\circ}\times\frac12\times\frac{1}{\sin75^\circ}$
$=\frac{1}{2}$
$=\text{R.H.S.}$
$=\cos(90^\circ-75^\circ)\cos(90^\circ-55^\circ)\frac{1}{\sin55^\circ}\times\frac12\times\frac{1}{\sin75^\circ}$
$=\sin75^\circ\sin55^\circ\frac{1}{\sin55^\circ}\times\frac12\times\frac{1}{\sin75^\circ}$
$=\frac{1}{2}$
$=\text{R.H.S.}$
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