Question
Evaluate the following:
$\big(\sin^230^\circ+4\cot^230-\frac34\sec^260^\circ\big)\big(\text{cosec}^245^\circ\sec^230^\circ\big)$
$\big(\sin^230^\circ+4\cot^230-\frac34\sec^260^\circ\big)\big(\text{cosec}^245^\circ\sec^230^\circ\big)$
✓
Answer
On substituting values of various T-ratious, we get:
$\big(\sin^230^\circ+4\cot^230-\frac34\sec^260^\circ\big)\\\big(\text{cosec}^245^\circ\sec^230^\circ\big)$
$=\Big[\Big(\frac12\Big)^2+4\times(1)^2-(2)^2\Big]\bigg[\Big(\sqrt{2}\Big)^2\Big(\frac{2}{\sqrt{3}}\Big)^2\bigg]$
$=\Big(\frac14+4-4\Big)\Big(2\times\frac43\Big)$
$=\frac14\times\frac83=\frac23$
$\big(\sin^230^\circ+4\cot^230-\frac34\sec^260^\circ\big)\\\big(\text{cosec}^245^\circ\sec^230^\circ\big)$
$=\Big[\Big(\frac12\Big)^2+4\times(1)^2-(2)^2\Big]\bigg[\Big(\sqrt{2}\Big)^2\Big(\frac{2}{\sqrt{3}}\Big)^2\bigg]$
$=\Big(\frac14+4-4\Big)\Big(2\times\frac43\Big)$
$=\frac14\times\frac83=\frac23$
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