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2 cos ? tan ? + tan ? = 1 + 2 cos ?

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Solution

2 cos tan + tan = 1 + 2 cos

tan(1 + 2 cos ) = 1 + 2 cos

(tan -1)(1 + 2 cos ) =0

tan =1 ----> = tan^-1(1) = k*pi + pi/4, k*pi +5pi/4     ( tan is +ve in Ist and IIrd quadrant)

1+2cos =0

cos = -1/2 ----> = pi-pi/3 , pi+pi/3                            ( cos is -ve in IInd and IIIrd quadrant)

= 2pi/3 +2k*pi , 4pi/3 +2k*pi

So, all solutions: = k*pi + pi/4, k*pi +5pi/4 , 2pi/3 +2k*pi , 4pi/3 +2k*pi

(b) In tnterval [0, 2pi)

= pi/4, 5pi/4 , 2pi/3 , 4pi/3