The Double Angle Identities

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These problems are for practicing the use of the double angle identities.

1 : $\alpha , \beta , \text { and } \gamma$ are angles in quadrants 1, 2, and 3, respectively, such that: $\begin{align*} \tan \alpha &= \sage{DispAlpha} \\ \tan \beta &= \sage{DispBeta} \\ \cos \gamma &= \sage{DispGamma} \end{align*}$

Assume also that $0 < \alpha , \beta , \gamma < 2\pi$ . Evaluate the following.

(a): $\sin (2\gamma ) = \answer {\sage {Ans1a}}$
(b): $\cos (2\beta ) = \answer {\sage {Ans1b}}$
(c): $\tan (2\alpha ) =\answer {\sage {Ans1c}}$
(d): $\csc (2\beta ) = \answer {\sage {Ans1d}}$
(e): $\sec (2\gamma ) = \answer {\sage {Ans1e}}$
(f): $\sin (4\gamma ) = \answer {\sage {Ans1f}}$
(g): $\cos (2\gamma +2\beta ) = \answer {\sage {Ans1g}}$

2 : Write the following as algebraic expressions.

(a): $\sin (2\arcsin (\sage {O2})) = \answer {\sage {Ans2a}}$
(b): $\tan (2\arcsin (\sage {O2})) = \answer {\sage {Ans2b}}$

3 : Find all values of $x$ satisfying the equation. I suggest not using the half angle formula. $\sin (\sage {Disp31}) + \cos (\sage {Disp32}) = 0$ $x = \begin {cases} \answer {\sage {Ans311}} + \answer {\sage {Ans312}} \cdot n \\ \answer {\sage {Ans321}} + \answer {\sage {Ans322}} \cdot n \\ \answer {\sage {Ans331}} + \answer {\sage {Ans332}} \cdot n \end {cases}$

4 : Determine whether each equality is true or false.

(a): $\sin (2x) = \frac {2\tan (x)}{1+\tan (x)^2}$
True False
(b): $\tan (2x) = \frac {2\sin (x)\cos (x)}{1+2\cos (x)^2}$
True False
(c): $\cos (2x) = \frac {1-\tan ^2(x)}{1+\tan (x)^2}$
True False
(d): $\sin (16x) = 16\sin (x)\cos (x)\cos (2x)\cos (4x)\cos (8x)$
True False