Question

tan thetha / 1 - cot thetha + cot theta /1- tan theta = 1 + sectheha × cosectheta​

Answer 1

Answer:

The ant taunted him to go and dance during the winter also. In conclusion it can be said that we must be responsible for our future. We must not just spend our good times idly; we must also secure our future against the bad times also.

Answer 2

Formula Used:-

• tanθ = 1/Cotθ
• tanθ = sinθ/cosθ
• cotθ = cosθ/sinθ
• secθ = 1/cosθ
• cosecθ = 1/sinθ
• sin²θ + cos²θ = 1
• a³ - b³ = (a - b)(a² + ab + b²)

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Solution:-

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$\tt{L.H.S. = \frac{tanθ}{1-cotθ} + \frac{cotθ}{1-tanθ}}$

$\tt{= \frac{tanθ}{1-\frac{1}{tanθ}} + \frac{\frac{1}{tanθ}}{1-tanθ}}$

$\tt{= \frac{tan²θ}{tan1-1} + \frac{1}{tanθ(1-tanθ)}}$

$\tt{= \frac{tan³θ-1}{tanθ(tanθ-1)}}$

$\tt{= \frac{\cancel{(tanθ-1)}(tan²θ+tanθ+1)}{tanθ\cancel{(tanθ-1)}}}$

$\tt{=\frac{tan²θ}{tanθ}+\frac{tanθ}{tanθ}+\frac{1}{tanθ}}$

$\tt{=tanθ + 1 + cotθ}$

$\tt{=1+tanθ + cotθ}$

$\tt{=1+ \frac{sinθ}{cosθ}+ \frac{cosθ}{sinθ}}$

$\tt{=1 + \frac{sin²θ + cos²θ}{Cosθ × Sinθ}}$

$\tt{=1+ \frac{1}{cosθ} × \frac{1}{sinθ}}$

$\tt{=1 + secθ × cosecθ}$

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$\tt{R.H.S. = 1 + secθ × cosecθ}$

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Hence,L.H.S. = R.H.S.

Proved✔