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If $\sin\theta+\sin^{2}\theta=1$ , then the value of $\cos^{12}\theta+3\cos^{10}\theta+3\cos^{8}\theta+\cos^{6}\theta-1$ is equal to

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1

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Trigonometric Identities -

$\sin ^{2}\Theta + \cos ^{2}\Theta = 1$

$1 + \tan ^{2}\Theta = \sec ^{2}\Theta$

$1 + \cot ^{2}\Theta = cosec ^{2}\Theta$

- wherein

They are true for all real values of $\Theta$

$\sin \theta +\sin ^{2}\theta = 1$

$\cos ^{12}\theta +3\cos ^{10}\theta +3\cos ^{8}\theta +\cos ^{6}\theta -1 =$

$\Rightarrow 1-\sin ^{2}\theta =\sin \theta$

$\cos ^{2}\theta =\sin \theta$

$\therefore \cos ^{12}\theta = \sin ^{6}\theta$

$\cos ^{10}\theta = \sin ^{5}\theta$

$\cos ^{8}\theta = \sin ^{4}\theta$

$\cos ^{6}\theta =\sin ^{3}\theta$

$\Rightarrow \sin ^{6}\theta +3\sin ^{5}\theta +3\sin ^{4}\theta +\sin ^{3}\theta -1$

$\Rightarrow \left ( \sin ^{2} \Theta \right )^{3} + \left ( \sin \theta \right )^{3} + 3 \sin \theta \cdot \sin ^{2}\theta (\sin \theta +\sin ^{2}\theta )-1$

$= \left ( 1 \right )^{3}-1\Rightarrow 1-1=0$

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