(ctgx+3)/(tg(x+pi/6))=ctg5pi/6

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To solve this equation, we can first simplify the expression on the left side of the equation:

(ctgx + 3) / tg(x + π/6) = ctg(5π/6)

Now let's express everything in terms of sin and cos:

(ctgx + 3) / (sin(x + π/6) / cos(x + π/6)) = cos(5π/6) / sin(5π/6)

Now, let's simplify the left side of the equation using the trigonometric identities:

(ctgx + 3) / (sin(x)cos(π/6) + cos(x)sin(π/6)) = cos(5π/6) / sin(5π/6)

(ctgx + 3) / (sin(x)(√3/2) + cos(x)(1/2)) = -√3/2 / 1/2

(ctgx + 3) / ((√3/2)sin(x) + (1/2)cos(x)) = -√3

Now, rewrite the left side of the equation using the definition of cotangent:

(1/tanx + 3) / ((√3/2)sin(x) + (1/2)cos(x)) = -√3

Now, we can multiply both sides by ((√3/2)sin(x) + (1/2)cos(x)) to get rid of the denominator:

1/tanx + 3 = -√3((√3/2)sin(x) + (1/2)*cos(x))

Now, continue simplifying the equation by re-expressing everything in terms of sine and cosine functions, using the definition of tangent function:

cos(x)/sin(x) + 3 = -√3(√3/2sin(x) + 1/2*cos(x))

cos(x)/sin(x) + 3 = -3/2sin(x) - √3/2cos(x)

Now, we can simplify the equation further by multiplying both sides by sin(x) to get rid of the denominator:

cos(x) + 3sin(x) = -3/2sin^2(x) - √3/2sin(x)cos(x)

This equation can be further simplified by rearranging terms and combining like terms to simplify to a single trigonometric function.