Sin альфа +2 sin2альфа+sin3aлфа=tg2 альфа/cosальфа+ 2cosa+cos3альфа

Предыдущий
вопрос Следующий
вопрос

To simplify the given expression, we can first rewrite it using the trigonometric identities for sine and cosine of sums:

sin α + 2sin^2 α + sin 3α = tg^2 α / cos α + 2cos α + cos 3α
sin α + 2sin^2 α + sin(2α + α) = tan^2 α / cos α + 2cos α + cos(2α + α)
sin α + 2sin^2 α + (sin 2α cos α + cos 2α sin α) = tan^2 α / cos α + 2cos α + (cos 2α cos α - sin 2α sin α)

Next, we can replace sin 2α and cos 2α using the double angle trigonometric identities:

sin α + 2sin^2 α + (2sin α cos α cos α + (1 - 2sin^2 α) sin α) = tan^2 α / cos α + 2cos α + ((2cos^2 α - 1) cos α - 2sin α * cos α)

sin α + 2sin^2 α + 2sin α cos^2 α + sin α - 2sin^2 α sin α = tan^2 α / cos α + 2cos α + (2cos^3 α - cos α - 2sin α * cos α)

Now we simplify further:

2sin α cos^2 α + sin α = tan^2 α / cos α + 2cos α + 2cos^3 α - cos α - 2sin α cos α

Finally, we can write the expression in terms of only α:

2sin α cos^2 α + sin α = sin α / cos α + 2cos α + 2cos^3 α - cos α - 2sin α cos α

This is the simplified form of the given expression.