If - - -12- prove that cos- - i sin-cos 2- - i sin 2-cos 3- - i sin 3- - 1

Now, (cosα + i sinα)(cos 2α + i sin 2α)cos 3α + i sin 3α =(cosα .cos 2α sinα.sin 2α)+ i( sinα.cos 2α+sin 2α.cosα)cos 3α + i sin 3α =(cos 3α + i si ...

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Byju's Answer

Standard XII

Mathematics

Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

If α = π12,...

Question

If $α = \frac{π}{12}$ , prove that $\frac{(cos α + i sin α) (cos 2 α + i sin 2 α)}{cos 3 α + i sin 3 α} = 1$

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Solution

Now,
$\frac{(cos α + i sin α) (cos 2 α + i sin 2 α)}{cos 3 α + i sin 3 α}$
$= \frac{(cos α . cos 2 α - sin α . sin 2 α) + i (sin α . cos 2 α + sin 2 α . cos α)}{cos 3 α + i sin 3 α}$
$= \frac{(cos 3 α + i sin 3 α)}{cos 3 α + i sin 3 α}$
$= 1$ .

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If α=π12, prove that (cosα+isinα)(cos2α+isin2α)cos3α+isin3α=1

Now,(cosα+isinα)(cos2α+isin2α)cos3α+isin3α=(cosα.cos2α−sinα.sin2α)+i(sinα.cos2α+sin2α.cosα)cos3α+isin3α=(cos3α+isin3α)cos3α+isin3α=1.

If $α = \frac{π}{12}$ , prove that $\frac{(cos α + i sin α) (cos 2 α + i sin 2 α)}{cos 3 α + i sin 3 α} = 1$

Now,
$\frac{(cos α + i sin α) (cos 2 α + i sin 2 α)}{cos 3 α + i sin 3 α}$
$= \frac{(cos α . cos 2 α - sin α . sin 2 α) + i (sin α . cos 2 α + sin 2 α . cos α)}{cos 3 α + i sin 3 α}$
$= \frac{(cos 3 α + i sin 3 α)}{cos 3 α + i sin 3 α}$
$= 1$ .