Trigonometric identity:

**sin θ / (1 - cos θ) = cosec θ + cot θ**The LHS of this identity does not seem to accept any trigonometric identities that we know, like sin

^{2}θ + cos

^{2}θ = 1, or tan θ = sin θ / cos θ. One way to solve trigonometric expressions containing rational expressions is to multiply the rational expression with the conjugate of the denominator.

The denominator of the LHS is (1 - cos θ), and its conjugate is (1 + cos θ). Multiplying by (1 + cos θ) / (1 + cos θ) is same as multiplying by 1, which makes no difference to an expression. So multiplying the LHS with (1 + cos θ),

sin θ / (1 - cos θ) * [(1 + cos θ) / (1 + cos θ)]

= [ sin θ (1 + cos θ) ] / [ (1 - cos θ) (1 + cos θ) ]The denominator is in the form of (a - b)(a + b), which can be simplified to a

^{2}- b

^{2},

= [ sin θ (1 + cos θ) ] / [ 1 - cosFrom the identity sin^{2}θ ]

^{2}θ + cos

^{2}θ = 1, it follows that sin

^{2}θ = 1 - cos

^{2}θ, putting this in the above expression,

= [ sin θ (1 + cos θ) ] / [ sinCancelling the common sin θ from the numerator and denominator,^{2}θ ]

= [ (1 + cos θ) ] / [ sin θ ]Distributing the denominator sin &theta in the numerator,

= 1/sin θ + cos θ / sin θRecall cosec θ = 1/sin θ and cot θ = cos θ / sin θ,

= cosec θ + cot θ... which is the RHS expression

Trigonometric identities applied:

- sin
^{2}θ + cos^{2}θ = 1 - cosec θ = 1/sin θ
- cot θ = cos θ / sin θ