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Sum and Difference Trigonometric Formulas

Product to Sum Trigonometric Formulas


If \[\sin 14^{\circ} \cos 56^{\circ} - \sin42^{\circ} \cos 28^{\circ} = \frac{1}{2}(K - \sin 14^{\circ}),\] what is the value of \(K?\)

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 \(\displaystyle{\frac{\tan x}{\tan y}}?\)

The minimum positive value of \(x\) that satisfies the equation \[\cos5x\cos2x=\sin2x\sin5x\] is \(\frac{\pi}{m},\) where \(m\) is a positive integer. What is \(m?\)


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