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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