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I am stuck in the following problems. Please help me. I was not sure what should be the appropriate title of this question. If you feel inadequate please edit it.

Q(1) If $x=\sin(\alpha-\beta)\sin(\gamma-\delta), y=\sin(\beta-\gamma)\sin(\alpha-\delta), z=\sin(\gamma-\alpha)\sin(\beta-\delta)$ then which one is/are true ?

(A)$x+y+z=0$

(B)$x^3+y^3+z^3=3xyz$

(C)$x+y-z=0$

(D)$x^3+y^3-z^3=3xyz$

Q(2) If $3\sin\beta=\sin(2\alpha+\beta)$ then which one is/are true ?

(A)$(\cot\alpha+\cot(\alpha+\beta))(\cot\beta-3\cot(2\alpha+\beta))=6$

(B)$\sin\beta=\cos(\alpha+\beta)\sin\alpha$

(C)$\tan(\alpha+\beta)=2\tan\alpha$

(D)$2\sin\beta=\sin(\alpha+\beta)\cos\alpha$

KON3
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2 Answers2

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HINT:

Use Prosthaphaeresis Formula, $2\sin A\sin B=\cos(A-B)-\cos(A+B)$

and $\cos(-x)=+\cos x$

For example, the first part of $2\sin(\alpha-\beta)\sin(\gamma-\delta)$, $\cos(\alpha-\beta-\gamma+\delta)$

and the last part of $2\sin(\gamma-\alpha)\sin(\beta-\delta)$ will be

$\cos(-\alpha+\beta+\gamma-\delta)=\cos\{-(\alpha-\beta-\gamma+\delta)\}=\cos(\alpha-\beta-\gamma+\delta)$

So, $(A)$ is correct which implies $(B)$

See If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.

Community
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lab bhattacharjee
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Can you see the cyclic rotation of angles so if you know the result for cyclic determinant its option B as for cyclic determinant $x^3+y^3+z^3=3xyz$ now its a special case for $x=y=z=0,1$ thus option $1$ is also true tgus options A,B are true.

Archis Welankar
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