1. Prove the following trigonometric identity. sin ( ) [ 1 + tan ( ) ] = tan ( ) [ sin

1. Prove the following trigonometric identity.

sin ( θ ) [ 1 + tan ( θ ) ] = tan ( θ ) [ sin ( θ ) + cos ( θ ) ]

Hint: apply the quotient identity that relates the tangent to the sine and cosine.

2. Express the sum as a product. cos ( 3 x ) + cos ( 7 x )

Hint: cos ( A ) + cos ( B ) = 2 cos [ ( A + B ) / 2 ] cos [ ( A − B ) / 2 ]

3. Given the expression cos ( 135 ° − 60 ° ) , use a difference formula to determine the exact value of cos ( 75 ° ) . Hint: cos ( A − B ) = cos ( A ) cos ( B ) + sin ( A ) sin ( B )

4. Given the expression sin ( 2 · 30 ° ) , use a double - angle formula to determine an alternate expression for sin ( 60 ° ) .