Question

$\csc^{6} \alpha = \cot^{6} \alpha + 3 \cot^{2} \alpha \ \csc^{2} \alpha + 1$
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Answer 1

Topic :-

Trigonometry

To Proof :-

$\csc^6\alpha=\cot^6\alpha+3\cot^2\alpha\cdot \csc^2\alpha+1$

Proof :-

Solving RHS,

$\leadsto \cot^6\alpha+3\cot^2\alpha\cdot \csc^2\alpha+1$

$\leadsto(\cot^2\alpha)^3+3\cot^2\alpha\cdot (1+\cot^2\alpha)+(1)^3$

$(\because \csc^2\alpha=1+\cot^2\alpha)$

$\leadsto (\cot^2\alpha+1)^3$

$(\because a^3+3ab(a+b)+b^3=(a+b)^3)$

$\leadsto(\csc^2\alpha)^3$

$(\because \csc^2\alpha=1+\cot^2\alpha)$

$\leadsto \csc^6\alpha$

LHS

$\leadsto \csc^6\alpha$

We observe that LHS = RHS.

Hence, Proved !!

Additional Formulae :-

$\sin^2\alpha+\cos^2\alpha=1$

$1+\tan^2\alpha=\sec^2\alpha$

$\sin 2\alpha= 2\sin \alpha \cdot \cos \alpha$

$\cos 2\alpha= \cos^2\alpha-\sin^2\alpha$

$\sin 3\alpha =3\sin \alpha -4\sin^3\alpha$

$\cos 3\alpha =4\cos^3\alpha-3\cos \alpha$