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### Trigonometry identities - 4

Trigonometric identity:  tan2θ - [ 1 / cos2θ ] + 1 = 0

This identity is composed of only tangent and cosine functions. We know that tan θ = sin θ / cos θ, so that means this identity can be converted wholly to sine and cosine functions. On the other hand, we know that sec θ = 1/cos θ and 1 + tan2θ = sec2θ, that means this identity can be represented wholly in secant function. Let us try both methods.

Method 1: Convert the identity to sine and cosine functions
Applying tan θ = sin θ / cos θ,
LHS = [ sin2θ / cos2θ ] - [ 1 / cos2θ ] + 1
Adding the fractions,
= [ sin2θ - 1 +  cos2θ] / [ cos2θ ]
Applying sin2θ + cos2θ = 1,
=  [1 - 1] / [ cos2θ ]
= 0 / [ cos2θ ]
Zero divided by any quantity is zero,
= 0
... which is the RHS

Method 2: Converting to secant function
From the identity 1 + tan2θ = sec2θ, it follows that tan2θ = sec2θ - 1,
LHS =  [sec2θ - 1] - [ 1 / cos2θ ] + 1
Applying sec θ = 1 / cos θ, it implies that sec2θ = 1 / cos2θ
= [ sec2θ - 1 ] - sec2θ + 1
Simplifying,
= 0
... which is the RHS

Trigonometric identities used in this post: