Trigonometric identities - 9

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 sin²θ + cos²θ = 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 a² - b²,

= [ sin θ (1 + cos θ) ] / [ 1 - cos²θ ]

From the identity sin²θ + cos²θ = 1, it follows that sin²θ = 1 - cos²θ, putting this in the above expression,

= [ sin θ (1 + cos θ) ] / [ sin²θ ]

Cancelling the common sin θ from the numerator and denominator,

=  [ (1 + cos θ) ] / [ sin θ ]

Distributing the denominator sin &theta in the numerator,

= 1/sin θ + cos θ / sin θ

Recall cosec θ = 1/sin θ and cot θ = cos θ / sin θ,

=  cosec θ + cot θ

... which is the RHS expression

Trigonometric identities applied:

• sin²θ + cos²θ = 1
• cosec θ = 1/sin θ
• cot θ = cos θ / sin θ