First slide

Trigonometric Identities

Question

The expression $3 [\sin^{4} (\frac{3}{2} π - α) + \sin^{4} (3 π + α)] - 2 [\sin^{6} (\frac{1}{2} π + α) + \sin^{6} (5 π - α)]$ is equal to

Moderate

A

0
B

1
C

3
D

none of these

Solution

$3 [\sin^{4} (\frac{3}{2} π - α) + \sin^{4} (3 π + α)] - 2 [\sin^{6} (\frac{1}{2} π + α) + \sin^{6} (5 π - α)]$
$\begin{array}{l} = 3 (\cos^{4} α + \sin^{4} α) - 2 (\cos^{6} α + \sin^{6} α) \\ = 3 (1 - 2 \sin^{2} {αcos}^{2} α) - 2 [{(\sin^{2} α + \cos^{2} α)}^{3} - 3 \sin^{2} {αcos}^{2} α (\sin^{2} α + \cos^{2} α)] \end{array}$
$\begin{array}{l} = 3 (1 - 2 \sin^{2} {αcos}^{2} α) - 2 [1 - 3 \sin^{2} {αcos}^{2} α] \\ = 1 \end{array}$

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