Trigonometric identities - 12

Trigonometric identity: 
The left hand side of this identity contains sine, cosine and tangent, while the right hand side contains only sine and cosine; The simplest way to solve this identity is to convert every trigonometric ratio on the left hand side to sine and cosine.

Applying the identities [ tan θ = sin θ / cos θ ] and [ cot θ = cos θ / sin θ ] to the LHS,

Simplifying each rational expression,

Writing sin A - cos A = - (cos A - sin A),

The denominator of both the fractions above is same, cos A - sin A. Adding the two fractions,

Applying the formula a2 - b2 = (a + b)(a - b),

Canceling the common factor (cos A - sin A) from the numerator and denominator,
= cos A + sin A
... which is the RHS expression

Trigonometric identities applied:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ