Question

Number of solutions of equation tan2x - 3tanx + cot2x - 3cot x + 4 = 0 in (-50pi, 50pi] ​

Answer 1

Concept-

Convert tan and cot in terms of sin and cos.

Given-

The equation is given as tan 2x- 3tanx + cot 2x - 3cotx +4 =0

Find-

Find the number of solutions of the given equation.

Solution-

tan2x-3tanx+cot2x-3cotx+4=0

tan2x+cot2x-3tanx-3cotx+4=0

(sin2x/cos2x +cos2x/sin2x)-3(sin x/cos x + cos x/sin x)+4=0

(sin²2x+cos²2x/ sin2xcos2x) - 3(sin²x+cos²x/sin x cos x)+4=0

(1/sin2xcos2x - 3/sin x cos x) +4=0

(2/sin4x - 3/ sin2x) +4 =0

(-1/sin4x - 3/sin2x) +2 =0

sin2x-3sin4x+2sin2xsin4x+0

sin2x[1-6cos2x+2sin4x]=0

sin2x=0 ⇒sin2x=sin nπ

⇒2x=nπ

x=nπ/2

So, the number of solutions of equation tan2x-3tanx+cot2x-3cotx=4=0 is nπ/2.