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# Sum and Difference Trigonometric Formulas

These formulas explain how to add and subtract trigonometric functions (and their arguments). If you've got sum time, see what a difference these formulas will make for your trig toolkit.

# 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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