Taking the left hand side expression of the given identity,
(1 + cot θ - cosec θ)(1 + tan θ + sec θ)Applying the following identities
- cot θ = cos θ / sin θ
- cosec θ = 1 / sin θ
- tan θ = sin θ / cos θ
- sec θ = 1 / cos θ
(1 + [cos θ / sin θ] - [1 / sin θ ])(1 + [sin θ / cos θ] + [1 / cos θ])Simplifying the fractions by taking a common denominator,
([sin θ + cos θ - 1] / sin θ)([cos θ + sin θ + 1] / cos θ)Multiplying the two rational expressions by multiplying together their numerators and denominator,
[(sin θ + cos θ - 1)(cos θ + sin θ + 1)] / [(sin θ)(cos θ)]Let a = sin θ + cos θ, then the fraction becomes
[(a - 1)(a + 1)] / [(sin θ)(cos θ)]Applying identity a2 - b2 = (a + b)(a - b),
[a2 - 12] / [(sin θ)(cos θ)]Put back a = sin θ + cos θ,
[(sin θ + cos θ)2 - 12] / [(sin θ)(cos θ)]Simplifying the numerator,
[sin2θ + cos2θ + 2(sinθ)(cosθ) - 1] / [(sin θ)(cos θ)]Applying identity sin2θ + cos2θ = 1,
[1 + 2(sinθ)(cosθ) - 1] / [(sin θ)(cos θ)]Simplifying the numerator,
[2(sinθ)(cosθ)] / [(sinθ)(cosθ)]Cancelling common (sinθ)(cosθ) from the numerator and denominator,
= 2... which is the RHS expression.
Identities applied:
- cot θ = cos θ / sin θ
- cosec θ = 1 / sin θ
- tan θ = sin θ / cos θ
- sec θ = 1 / cos θ
- sin2θ + cos2θ = 1
