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Let theta= a for typing convenience.
sin a tan a/ 1-cos a
Multiplying numerator and denominator by cos a
sin a* sin a/cos a* cos a / cos a(1-cos a)
=sin²a/ cos a(1- cos a)
= 1-cos²a/cos a(1-cos a)
= (1+cos a)(1-cos a)/cos a(1-cos a)
= 1+cos a/cos a
= 1+sec a------------------------proved
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