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Monica [59]
2 years ago
9

Question is attached.Step by step explanation only!​

Mathematics
2 answers:
olchik [2.2K]2 years ago
8 0

Given :

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

To Do :-

  • To prove LHS = RHS

Proof :-

<u>We </u><u>know</u><u> that</u><u> </u><u>,</u>

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

<u>On </u><u>using </u><u>above</u><u> </u><u>two </u><u>in </u><u>LHS</u><u> </u><u>,</u>

  • 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 !

il63 [147K]2 years ago
5 0

Answer:

See Below.

Step-by-step explanation:

We want to verify the equation:

\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}}

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

\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}

Hence verified.

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