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If $x$ satisfies the equation $$\tan x = \dfrac{\sin 10^\circ + \sin 40^\circ}{ \cos 10^\circ + \cos 40^\circ}$$ and $x$ is between $0^\circ$ and $90^\circ$, then $x$ is equal to what?

Is there an identity I can use here?

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closed as off-topic by Noah Schweber, астон вілла олоф мэллбэрг, Stefan Mesken, Daniel W. Farlow, user26857 Nov 13 '16 at 20:53

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Try this in the numerator:

$$\sin a + \sin b = 2 \sin \frac{a+b}2 \;\cos \frac{a-b}2$$

And this in the denominator:

$$\cos a + \cos b = 2 \cos \frac{a+b}2\;\cos\frac{a-b}2$$

Divide the numerator by the denominator after applying these sum-product relations, cancel out common factors, and see what you get.

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  • $\begingroup$ Just enough hints to stimulate curiosity! +1. $\endgroup$ – blackpen Nov 13 '16 at 4:44
  • $\begingroup$ (+1) The identity that ensues is one of my favorites. It also shows that $\tan\left(\frac a2\right)=\frac{\sin(a)}{1+\cos(a)}$ $\endgroup$ – robjohn Nov 13 '16 at 4:56
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Proof Without Words

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$$ \large\color{#8060A0}{\frac{\sin(a)+\sin(b)}{\cos(a)+\cos(b)}=\tan\left(\frac{a+b}2\right)} $$

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