Simplify 2sin(5x)cos(3x) - sin(2x) to one an expression containing one trigonometric function. Then graph the original function and your simplified version to verify they are identical. Enclose arguments of functions in parentheses. For example, sin(2x). 

2sin(5x)cos(3x)-sin(2x)=

In triangle \(A B C\) , if altitude \(A D\) is drawn to side \(B C\) , which of the following must be true? 

(1.) \(\angle ADB \cong \angle A D C\) 

2. \(\overline { B D } \cong \overline { D C } \) 

3. \(\triangle A D B \cong \triangle A D C\) 

4. \(\angle B \cong \angle C\) 

1) Use the limit definition of the derivative \(( \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } ) \) together with the tangent addition formula to prove that \(\frac { d } { d x } [ \tan x ] = \sec ^ { 2 } x\) 

Tangent Addition Formula: \(\tan ( x + y ) = \frac { \tan x + \tan y } { 1 - \tan x \tan y } \) 

You may also use this graph: \(f ( x ) = \frac { \tan x } { x } \) 

Find all the solutions to \(\displaystyle \left ( { 2t - 3} \right ) \tan \left ( { \frac { { 6t} } { { 11} } } \right ) = 15 - 10t\). Use at least 4 decimal places in your work.

Find the exact value of each expression, if it exists. 

arcsin \( ( - \frac { 1 } { 2 } ) \) 

a. \( \frac { \pi } { 6 } \) 

b. \( - \frac { \pi } { 6 } \) 

c. \( - \frac { 1 } { 2 } \) 

d. undefined 

Find the exact value of the sine function of the given angle. 

\( 690 ^ { \circ } \) 

\( \sin 690 ^ { \circ } = \) 

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)