GIVEN:

\(\frac{{tan\theta \left( {1 - sin\theta } \right)\left( {sec\theta \; + \;tan\theta } \right)}}{{\left( {1\; + \;cos\theta } \right)\left( {cosec\theta - cot\theta } \right)}}\)

FORMULA USED:

1 – sin²θ = cos²θ; 1 – cos²θ = sin²θ

(a + b)(a – b) = a² – b²

CALCULATION:

\(\frac{{tan\theta \left( {1 - sin\theta } \right)\left( {sec\theta \; + \;tan\theta } \right)}}{{\left( {1\; + \;cos\theta } \right)\left( {cosec\theta - cot\theta } \right)}}\)

⇒ \(\frac{{tan\theta \left( {1 - sin\theta } \right)\left( {\frac{1}{{cos\theta }}\; + \;\frac{{sin\theta }}{{cos\theta }}} \right)}}{{\left( {1\; + \;cos\theta } \right)\left( {\frac{1}{{sin\theta }}\; - \;\frac{{cos\theta }}{{sin\theta }}} \right)}}\)

⇒ \(\frac{{tan\theta \left( {1 - sin\theta } \right)\left( {1\; + \;sin\theta } \right)sin\theta }}{{\left( {1\; + \;cos\theta } \right)\left( {1 - cos\theta } \right)cos\theta }}\)

⇒ \(\frac{{tan\theta \left( {1 - {{\sin }^2}\theta } \right)}}{{\left( {1 - {{\cos }^2}\theta } \right)}}\; \times \;tan\theta \)

⇒ \(\frac{{tan\theta \left( {{{\cos }^2}\theta } \right)}}{{{{\sin }^2}\theta }}\; \times \;tan\theta \)

⇒ tan²θ × cot²θ

⇒ 1