.Express `cos 6 alpha ` in terms of ` cos 3 alpha `.

(cos alpha + 2 cos 3 alpha + cos 5 alpha )/(cos 3 alpha + 2 cos 5 alpha + cos 7 alpha ) = cos 3 alpha sec 5 alpha .

If 10sin^4alpha+15cos^4alpha=6 , find the value of 27cos e c^6alpha+8s e c^6alpha .

Express 2cos theta + 3 sin theta in the form r cos ( theta - alpha ), where r is positive, stating the values of r and alpha. What is the maximum value of 2 cos theta + 3 sin theta ?

cos alpha cos(60^(@) - alpha ) cos (60^(@) + alpha ) = 1/4 cos 3 alpha.

If alpha+gamma=2beta then the expression (sin alpha-sin gamma)/(cos gamma-cos alpha) simplifies to:

Express the following in a + ib form: ((cos alpha + i sin alpha)^(4))/((sin beta + i cos beta)^(5))

If 10sin^4 alpha+15 cos^4 alpha=6, then find the value of 27 cosec^6 alpha+8sec^6 alpha.

If alpha and beta are acute angles such that alpha+beta=lamda , where constant, find the maximum possible value of the expression sin alpha+sin beta+cos alpha+cos beta .

Prove that (sin alpha cos beta + cos alpha sin beta) ^(2) + (cos alpha coa beta - sin alpha sin beta) ^(2) =1.

Prove that the equation of a line passes through the point (a cos^(3)alpha , a sin^(3)alpha ) and perpendicular to the line x tan alpha+y=a sin alpha is x cos alpha-y sin alpha-a cos 2 alpha .