Trigonometric identity:

**(1 + tan**^{2}θ)(1 - sin θ)(1 +sin θ) = 1This trigonometric identity can, as the previous one, be solved in two ways:

- Convert all trigonometric functions to sine and cosine and simplify
- Apply identities other than that (more advanced method)

Recall the trigonometric identity 1 + tan

^{2}θ = sec

^{2}θ. Applying it to the LHS expression,

(1 + tan^{2}θ)(1 - sin θ)(1 +sin θ)

= (secThe 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 a^{2}θ)(1 - sin θ)(1 +sin θ)

^{2}- b

^{2}, so it can be simplified to 1

^{2}- sin

^{2}θ,

= (secFrom the identity sin^{2}θ)(1 - sin^{2}θ)

^{2}θ + cos

^{2}θ = 1, it follows that cos

^{2}θ = 1 - sin

^{2}θ, applying this to the above expression,

= (secWe know that sec θ = 1/cos θ, hence sec^{2}θ)(cos^{2}θ)

^{2}θ = 1/cos

^{2}θ. Hence the expression becomes,

= [ 1/cos^{2}θ ] * cos^{2},

= 1... which is same as the RHS.

Trigonometric identities applied:

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