1. Derive formulas for the following:

a. sin (2theta)

b. cos (2theta)

Sin (2theta) = 2sintheta costheta

Cos (2theta) = 2cos²theta - 1

Cos (2theta) = 1 - 2sin²theta

Step-by-step explanation:

Sin (2theta) = Sin (theta+theta)

Sin (theta+theta) = sintheta costheta + costheta sintheta

Since costheta and sintheta are occurring twice in the addition

Sin (theta+theta) = 2sintheta costheta

Cos (2theta) = cos (theta+theta)

= costheta costheta - sintheta sintheta

= cos²theta - sin²theta ... 1

From sin²theta + cos²theta = 1

Sin²theta = 1 - cos²theta ... 2

Substituting 2 into 1

Cos (2theta) = cos²theta - (1 - cos²theta)

Cos (2theta) = cos²theta - 1 + cos²theta

Cos (2theta) = 2cos²theta - 1

Cos (2theta) = 1 - 2sin²theta