Geometry
Sum and Difference Trigonometric Formulas
If \[ a \neq 0, \sin(x+y) = \frac{11}{a}, \sin(x-y) = \frac{2}{a},\] what is \(\displaystyle{\frac{\tan x}{\tan y}}?\)
If \(\cos \alpha=\frac{1}{11}\) and \(\cos \beta=\frac{10}{11}\), the value of \[\sin (\alpha+\beta) \sin (\alpha-\beta)\] can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?
\(N\) is an integer in the range \( [0, 180] \) such that
\[ 2 \cos 43 ^\circ \cos 26 ^\circ = \cos 69 ^ \circ + \cos N ^ \circ. \]
What is the value of \(N\)?
If \[ a \neq 0, \sin(x+y) = \frac{12}{a}, \sin(x-y) = \frac{5}{a},\] what is \(\displaystyle{\frac{\tan x}{\tan y}}?\)
Which of the following is equal to \( \cos 6 \theta \cos 2 \theta?\)