Trigonometric identities - 7

Trigonometric identity:  (1 + tan2θ)(1 - sin θ)(1 +sin θ) = 1

This trigonometric identity can, as the previous one, be solved in two ways:
  1. Convert all trigonometric functions to sine and cosine and simplify
  2. Apply identities other than that (more advanced method)
The latter is more advanced because it requires you to remember the various trigonometric identities (actually, only the first three ones) and apply them at correct places. Let us solve with the latter method.

Recall the trigonometric identity 1 + tan2θ = sec2θ. Applying it to the LHS expression,
(1 + tan2θ)(1 - sin θ)(1 +sin θ)
= (sec2θ)(1 - sin θ)(1 +sin θ)
The other part of the LHS, (1 - sin θ)(1 +sin θ), is in the form of (a + b)(a - b), which can be simplified to a2 - b2, so it can be simplified to 12 - sin2 θ,
=  (sec2θ)(1 - sin2 θ)
From the identity sin2 θ + cos2 θ = 1, it follows that cos2 θ = 1 - sin2 θ, applying this to the above expression,
=  (sec2θ)(cos2 θ)
We know that sec θ = 1/cos θ, hence sec2θ = 1/cos2θ. Hence the expression becomes,
= [ 1/cos2θ ] * cos2,
= 1
... which is same as the RHS.

Trigonometric identities applied:

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