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Mathematics

Prove that :

4 (sin4 30° + cos4 60°) - 3 (cos2 45° - sin2 90°) = 2

Trigonometric Identities

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Answer

4 (sin4 30° + cos4 60°) - 3 (cos2 45° - sin2 90°) = 2

L.H.S. = 4 (sin4 30° + cos4 60°) - 3 (cos2 45° - sin2 90°)

=4×((12)4+(12)4)−3×((12)2−(1)2)=4×(116+116)−3×(12−1)=4×(1+116)−3×(12−22)=4×(216)−3×(1−22)=4×(18)−3×(−12)=12+32=1+32=42=2= 4 \times \Big(\Big(\dfrac{1}{2}\Big)^4 + \Big(\dfrac{1}{2}\Big)^4\Big) - 3 \times \Big(\Big(\dfrac{1}{\sqrt2}\Big)^2 - (1)^2\Big)\\[1em] = 4 \times \Big(\dfrac{1}{16} + \dfrac{1}{16}\Big) - 3 \times \Big(\dfrac{1}{2} - 1\Big)\\[1em] = 4 \times \Big(\dfrac{1 + 1}{16}\Big) - 3 \times \Big(\dfrac{1}{2} - \dfrac{2}{2}\Big)\\[1em] = 4 \times \Big(\dfrac{2}{16}\Big) - 3 \times \Big(\dfrac{1 - 2}{2}\Big)\\[1em] = 4 \times \Big(\dfrac{1}{8}\Big) - 3 \times \Big(\dfrac{-1}{2}\Big)\\[1em] = \dfrac{1}{2} + \dfrac{3}{2}\\[1em] = \dfrac{1 + 3}{2}\\[1em] = \dfrac{ 4}{2}\\[1em] = 2

R.H.S. = 2

∴ L.H.S. = R.H.S.

Hence, 4 (sin4 30° + cos4 60°) - 3 (cos2 45° - sin2 90°) = 2.

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