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Question

If $$\sin \theta - \cos \theta = 1$$, then the value of $$\sin^3 \theta - \cos ^3 \theta $$ is equal to 


A
1
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B
1
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C
0
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D
2
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E
2
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Solution

The correct option is B $$1$$
Given, $$\sin\theta-\cos\theta=1$$
$$\sin^3\theta -\cos^3\theta=(\sin\theta-\cos\theta)(\sin^2\theta+\cos^2\theta+\sin\theta\cos\theta)$$
$$=1(1+\sin\theta\cos\theta)$$
$$=1+\sin\theta\cos\theta$$.........(i)
$$[\because \sin\theta-\cos\theta=1]$$
On squaring both sides,
$$(\sin\theta-\cos\theta)^2=(1)^2$$
$$\Rightarrow (\sin^2\theta+\cos^2\theta-2\sin\theta\cos\theta)=1$$
$$\Rightarrow 1-2\sin\theta\cos\theta=1$$
$$\sin\theta\cos\theta=0$$.........(ii)
From Equations (ii) and (i), we get $$\sin^3\theta-\cos^3=1$$

Mathematics

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