Question

Prove that tan o/1-cot o +cot o/1-tan o = 1+sec o.cosec o
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Answer 1

Answer:

1-cot o

=1-cos o/sin o

=(sin o -cos o)/sin o

similarly, 1-tan o =(cos o-sin o)/cos o

tan o/1-cot o +cot o/1-tan o

=(sin o/cos o) /(sin o -cos o)/sin o + (cos o/sin o) / (cos o- sin o)/ cos o

=sin²o /cos o(sin o-cos o) + cos²o/sin o(cos o-sin o)

=sin²o/(sin o.cos o- cos²o) + cos²o/(sin o cos o-sin²o)

=(sin³o.cos o-sin⁴o+sin o.cos³o-cos⁴o)/ (sin o.cos o-cos²o).(sin o.cos o-sin²o)

=sin o.cos o(sin²o+cos²o)-(sin⁴o+cos⁴o) / {(sin o.cos o)²-sin o.cos o(sin²o+cos²o)+(sin o.cos o)²}

={sin o.cos o-(sin²o+cos²o)²+2.sin²o.cos²o} / {2.sin²o.cos²o-(sin o.cos o)}

={sin o.cos o-(1)²+2.sin²o.cos²o}/{2.sin²o.cos²o-(sin o.cos o)}

={2.sin²o.cos²o-sin o.cos o+2.sin o.cos o-1}/{2.sin²o.cos²o-(sin o.cos o)}

={2.sin²o.cos²o-(sin o.cos o)}/{2.sin²o.cos²o-(sin o.cos o)} + (2.sin o.cos o -1)/{sin o.cos o(2.sin o.cos o-1)}

=1 + (1/sin o.cos o)

=1 + sec o.cosec o.....[proved]