<s>[INST] Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$ Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$
$$\tan x + \sec x = 2\cos x$$
I tried changing it all to sin and cos
$$\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2\cos x$$
then I made it to one fraction
$$\frac{\sin x + 1}{\cos x} = 2 \cos x$$
Then I don't know where to go from there. Please help!
 [/INST] $$\sec x+\tan x=2\cos x\iff \sec x-\tan x=\frac1{2\cos x}$$
Adding we get $$2\sec x=2\cos x+\frac1{2\cos x}$$
Multiplying by $2\cos x,$
$$4=4\cos^2x+1\iff2\cos^2x=\frac32\iff\cos2x=2\cos^2x-1=\frac12=\cos60^\circ$$
$$\iff 2x=360^\circ m\pm60^\circ$$
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