Respuesta :

xKelvin

Answer:

See Below.

Step-by-step explanation:

We want to verify the equation:

[tex]\displaystyle \frac{\tan\theta}{ 1- \cot \theta} + \frac{\cot\theta}{1 - \tan\theta} = \frac{\dfrac{\sin\theta}{\cos\theta}}{1 - \dfrac{\cos\theta}{\sin\theta}} + \frac{\dfrac{\cos\theta}{\sin\theta}}{1 - \dfrac{\sin\theta}{\cos\theta}}[/tex]

Recall that tanθ = sinθ / cosθ. Likewise, cotθ = cosθ / sinθ. Then by substitution:

[tex]\displaystyle \begin{aligned} \frac{\tan\theta}{ 1- \cot \theta} + \frac{\cot\theta}{1 - \tan\theta} & = \frac{\dfrac{\sin\theta}{\cos\theta}}{1 - \dfrac{\cos\theta}{\sin\theta}} + \frac{\dfrac{\cos\theta}{\sin\theta}}{1 - \dfrac{\sin\theta}{\cos\theta}} \\ \\ & \stackrel{\checkmark}{=} \frac{\dfrac{\sin\theta}{\cos\theta}}{1 - \dfrac{\cos\theta}{\sin\theta}} + \frac{\dfrac{\cos\theta}{\sin\theta}}{1 - \dfrac{\sin\theta}{\cos\theta}} \end{aligned}[/tex]

Hence verified.

VirαtKσhli

Given :

  • tan∅/1-cot∅ + cot∅/1-tan∅ = sin∅/cos∅ ÷ (1- cos∅/sin∅ ) + cos∅/sin∅ ÷ (1 - sin∅/cos∅)

To Do :-

  • To prove LHS = RHS

Proof :-

We know that ,

  • sin∅/cos∅ = tan∅
  • cos∅/sin∅ = cot∅

On using above two in LHS ,

  • LHS = tan∅/1-cot∅ + cot∅/1-tan∅
  • LHS = tan∅/1-cot∅ + cot∅/1-tan∅ = sin∅/cos∅ ÷ (1- cos∅/sin∅ ) + cos∅/sin∅ ÷ (1 - sin∅/cos∅)
  • LHS = RHS

Hence proved !

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