Question

(sin α + sin 5 α)/(cos α + cos 5 α)

Answer 1

(sin α + sin 5 α)= 2Sin3α Cos2α
(cos α + cos 5 α) = 2Cos3α Cos2α
делим и пишем ответ: tg3α

Answer 2

To simplify the expression (sin α + sin 5 α)/(cos α + cos 5 α), we can use the following trigonometric identity:

sin A + sin B = 2 sin ((A + B)/2) cos ((A - B)/2)

and

cos A + cos B = 2 cos ((A + B)/2) cos ((A - B)/2)

Using this identity, we can rewrite the numerator and denominator of the expression as follows:

sin α + sin 5 α = 2 sin ((α + 5α)/2) cos ((5α - α)/2) = 2 sin (3α) cos (2α)

cos α + cos 5 α = 2 cos ((α + 5α)/2) cos ((5α - α)/2) = 2 cos (3α) cos (2α)

Substituting these expressions back into the original expression, we get:

(sin α + sin 5 α)/(cos α + cos 5 α) = (2 sin (3α) cos (2α))/(2 cos (3α) cos (2α))

The common factor of 2 cos (2α) cancels out, leaving us with:

tan (3α)

Therefore, the simplified form of the expression is tan (3α).