The number of roots of the equation cos^7 the | vedclass

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(A) Given the equation: $\cos^7 	heta - \sin^4 	heta = 1$Rearranging the terms,we get: $\cos^7 	heta = 1 + \sin^4 	heta$We know that for any $	he

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$2$

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(A) Given the equation: $\cos^7 	heta - \sin^4 	heta = 1$Rearranging the terms,we get: $\cos^7 	heta = 1 + \sin^4 	heta$We know that for any $	heta$,the range of $\cos^7 	heta$ is $[-1, 1]$.Also,since $\sin^4 	heta \geq 0$,the right-hand side $(RHS)$ satisfies $1 + \sin^4 	heta \geq 1$.For the equality to hold,both sides must be equal to $1$.Thus,$\cos^7 	heta = 1$ and $\sin^4 	heta = 0$.From $\cos^7 	heta = 1$,we get $\cos 	heta = 1$,which implies $	heta = 0, 2\pi$ in the interval $[0, 2\pi]$.From $\sin^4 	heta = 0$,we get $\sin 	heta = 0$,which implies $	heta = 0, \pi, 2\pi$ in the interval $[0, 2\pi]$.The common values of $	heta$ satisfying both conditions are $	heta = 0$ and $	heta = 2\pi$.Therefore,there are $2$ roots in the given interval.

The number of roots of the equation $\cos^7 \theta - \sin^4 \theta = 1$ that lie in the interval $[0, 2\pi]$ is

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