First slide

Trigonometric Identities

Question

If $\sin α + \cos α = m,$ then $\sin^{6} α + \cos^{6} α$ is equal to

Moderate

A

$\frac{4 - 3 {(m^{2} - 1)}^{2}}{4}$
B

$\frac{4 + 3 {(m^{2} - 1)}^{2}}{4}$
C

$\frac{3 + 4 {(m^{2} - 1)}^{2}}{4}$
D

none of these

Solution

$\sin α + \cos α = m \Rightarrow 1 + \sin 2 α = m^{2} \Rightarrow \sin 2 α = m^{2} - 1$
Now,
$\begin{array}{l} \sin^{6} α + \cos^{6} α \\ = {(\sin^{2} α + \cos^{2} α)}^{3} - 3 \sin^{2} α \cos^{2} α (\sin^{2} α + \cos^{2} α) \\ = 1 - 3 \frac{{(- 1 + m^{2})}^{2}}{4} = \frac{4 - 3 {(m^{2} - 1)}^{2}}{4} \end{array}$

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