# Help me to solve the following. 1. Simplify `2cos ((5pi)/12) sin ((5pi)/12)` 2. Solve for x of (4 sin^2 x -1) (cos x - 2) = 0 where 0 ≤ x ≤ 2π 3. tan^2 β -3 = 0 4. Simplify sin(π/4 + 0) +...

Help me to solve the following.

1. Simplify `2cos ((5pi)/12) sin ((5pi)/12)`

2. Solve for x of (4 sin^2 x -1) (cos x - 2) = 0 where 0 ≤ x ≤ 2π

3. tan^2 β -3 = 0

4. Simplify sin(π/4 + 0) + sin(π/4 - 0)

5. Solve for θ; sin 40 = √3sin 20

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Expert Answers

lemjay | Certified Educator

1.) `2 cos ((5pi)/12)sin((5pi)/12)`

To simplify, apply the double angle identity of sine which is `sin(2theta)=2sin theta cos theta` .

`2 cos ((5pi)/12)sin((5pi)/12) = 2 sin((5pi)/12 )cos ((5pi)/12)`

`=sin (2*(5pi)/12)=sin((5pi)/6)`

And, refer to Unit Circle Chart to get the exact value of sine when the angle is (5pi)/6.

`=1/2`

**Hence, `2cos((5pi)/12)sin((5pi)/12)=1/2` .**

*(For the other four problems, please post them as separate questions is HOmework Help. )*