Solve the trigonometric equation $tan θ + cot θ = 2$ $tantheta+cottheta=2$ ?

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Answer 1 · 2017-10-08T15:23:55

$θ = n π + \frac{π}{4}$ $theta=npi+pi/4$

$tan θ + cot θ = 2$ $tantheta+cottheta=2$

i.e. $\frac{sin θ}{cos θ} + \frac{cos θ}{sin θ} = 2$ $sintheta/costheta+costheta/sintheta=2$

or $\frac{{sin}^{2} θ + {cos}^{2} θ}{sin θ cos θ} = 2$ $(sin^2theta+cos^2theta)/(sinthetacostheta)=2$

or $\frac{1}{sin θ cos θ} = 2$ $1/(sinthetacostheta)=2$

or $2 sin θ cos θ = 1$ $2sinthetacostheta=1$

or $sin 2 θ = sin (\frac{π}{2})$ $sin2theta=sin(pi/2)$

Hence $2 θ = 2 n π + \frac{π}{2}$ $2theta=2npi+pi/2$

or $θ = n π + \frac{π}{4}$ $theta=npi+pi/4$

Answer 2 · 2017-10-08T15:26:18

$tan θ + cot θ = 2$ $tantheta+cottheta=2$

$\Rightarrow tan θ + \frac{1}{tan θ} = 2$ $=>tantheta+1/tantheta=2$

$\Rightarrow {tan}^{2} θ - 2 tan θ + 1 = 0$ $=>tan^2theta-2tantheta+1=0$

$\Rightarrow {(tan θ - 1)}^{2} = 0$ $=>(tantheta-1)^2=0$

$\Rightarrow tan θ = 1 = tan (\frac{π}{4})$ $=>tantheta=1=tan(pi/4)$

$=>theta=npi+pi/4" where "n in ZZ$

Solve the trigonometric equation tantheta+cottheta=2tanθ+cotθ=2tantheta+cottheta=2?