Question : If $\cos \theta+\sec \theta=2$, then the value of $\sin ^6 \theta+\cos ^6 \theta$ is:
Option 1: $\frac{1}{3}$
Option 2: $0$
Option 3: $1$
Option 4: $\frac{1}{2}$
Correct Answer: $1$
Solution : Given that $\cos \theta + \sec \theta = 2$, $⇒\cos \theta + \frac{1}{\cos \theta} = 2$ $⇒\cos^2 \theta + 1 = 2\cos \theta$ $⇒\cos^2 \theta - 2\cos \theta + 1 = 0$ $⇒(\cos \theta - 1)^2 = 0$ $⇒\cos \theta = 1$ $⇒\theta=0°$ $⇒\sin \theta = \sin0° = 0$ So, $\sin^6 \theta + \cos^6 \theta = 0^6 + 1^6 = 0 + 1 = 1$ Hence, the correct answer is $1$.
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