First slide

Trigonometric Identities

Question

The value of $3 \frac{\sin^{4} t + \cos^{4} t - 1}{\sin^{6} t + \cos^{6} t - 1}$ is equal to

Moderate

Solution

$\begin{array}{l} \begin{aligned} \sin^{4} t + \cos^{4} t - 1 & = {(\sin^{2} t + \cos^{2} t)}^{2} - 2 \sin^{2} {tcos}^{2} t - 1 \\ = - 2 \sin^{2} {tcos}^{2} t \end{aligned} \\ \begin{aligned} \sin^{6} t + \cos^{6} t - 1 & = {(\sin^{2} t + \cos^{2} t)}^{3} - 3 \sin^{2} {tcos}^{2} t - 1 \\ = - 3 \sin^{2} {tcos}^{2} t \end{aligned} \\ Hence, 3 \frac{\sin^{4} t + \cos^{4} t - 1}{\sin^{6} t + \cos^{6} t - 1} = 2 \end{array}$

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