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Trigonometrical Identities — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsTrigonometrical Identities3 Marks
Question
Evaluate without using trigonometric tables,$\sin ^2 28^{\circ}+\sin ^2 62^{\circ}+\tan ^2 38^{\circ}-\cot ^2 52^{\circ}+\frac{1}{4} \sec ^2 30^{\circ}$
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Answer

$\sin ^2 28^{\circ}+\sin ^2 62^{\circ}+\tan ^2 38^{\circ}-\cot ^2 52^{\circ}+\frac{1}{4} \sec ^2 30^{\circ}$
$=\sin ^2 28^{\circ}+\sin ^2\left(90^{\circ}-28^{\circ}\right)+\tan ^2 38^{\circ}-\cot ^2\left(90^{\circ}-38^{\circ}\right)+\frac{1}{4} \sec ^2 30^{\circ}$
$=\left(\sin ^2 28^{\circ}+\cos ^2 28^{\circ}\right)+\tan ^2 38^{\circ}-\tan ^2 38^{\circ}+\frac{1}{4} \times\left(\frac{2}{\sqrt{3}}\right)^2$
$=1+0+\frac{1}{4} \times \frac{4}{3}$
$=1+\frac{1}{3}$
$=\frac{4}{3}$

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