Question

Prove that (sin^2 30°)* (cos^2 45°) +1/2 sin^2 90° + 1/8 cot^2 60° = 2

Answer 1

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$\underline{\bold{\: Prove \: \: That \: \: :}}$


$\bold{(sin {}^{2}30 \degree)(cos {}^{2} 45 \degree) + \frac{1}{2} \: sin {}^{2} 90 \degree + \frac{1}{8} \: cot {}^{2}60 \degree = \frac{2}{3} }$

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$\bold{L.H.S. = (sin {}^{2} 30\degree)(cos {}^{2}45 \degree) + \frac{1}{2} \: sin {}^{2}90 \degree + \frac{1}{8} \: cot {}^{2} 60 \degree } \\ \\ \bold{ = (\frac{1}{2} ) {}^{2} \times ( \frac{1}{ \sqrt{2} }) {}^{2} + \frac{1}{2} \times (1) {}^{2} + \frac{1}{8} \times (\frac{1}{ \sqrt{3} }) {}^{2} } \\ \\ \bold{ = \frac{1}{4} \times \frac{1}{2} + \frac{1}{2} + \frac{1}{8} \times \frac{1}{3} } \\ \\ \bold{ = \frac{1}{8} + \frac{1}{2} + \frac{1}{24} } \\ \\ \bold{ = \frac{3 + 12 + 1}{24} } \\ \\ \bold{ = \frac{16}{ 24} } \\ \\ \bold{ = \frac{2}{3} } \\ \\ \bold{ = R.H.S.} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{\bold{ \: \: Hence \: \: Proved. \: \: }}$