Question

4(sin⁴30°+cos⁴60)-3(cos²45°+sin²90°)​

Answer 1

$\green{\bold{\underline{ ☆ UPSC-ASPIRANT ☆} }}$

$\red{\bold{\underline{\underline{QUESTION:-}}}}$

4(sin⁴30°+cos⁴60)-3(cos²45°+sin²90°)

$\huge\bigstar\huge\tt\underline\red{ᴀɴsᴡᴇʀ }\bigstar$

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Answer 2

$\sf\huge\blue{\underline{\underline{ Question : }}}$

4(sin⁴ 30° + cos⁴ 60) - 3(cos² 45° + sin² 90°)

$\sf\huge\blue{\underline{\underline{ Solution : }}}$

$\rm\:\implies 4(\sin^{4}\:30\degree + \cos^{4}\:60\degree) - 3(\cos^{2}\:45\degree + \sin^{2}\:90\degree)$

• sin 30° = 1/2
• cos 60° = 1/2
• cos 45° = 1/√2
• sin 90° = 1

$\bf\:\implies 4((\frac{1}{2})^{4} + (\frac{1}{2})^{4}) - 3((\frac{1}{\sqrt{2}})^{2} + (1)^{2})$

$\bf\:\implies 4(\frac{1}{16} + \frac{1}{16} ) - 3(\frac{1}{2} + 1)$

$\bf\:\implies 4(\frac{1 + 1}{16} ) - 3(\frac{1 + 2}{2})$

$\bf\:\implies 4(\frac{2}{16} ) - 3(\frac{3}{2})$

$\bf\:\implies \frac{1}{2} - \frac{9}{2}$

$\bf\:\implies \frac{1 - 9}{2}$

$\bf\:\implies \frac{- 8}{2}$

$\bf\:\implies - 4$

$\underline{\boxed{\bf{\purple{\therefore4(\sin^{4}\:30\degree + \cos^{4}\:60\degree) - 3(\cos^{2}\:45\degree + \sin^{2}\:90\degree) = - 4 }}}}\:\orange{\bigstar}$