Trigonometric identity:

**tan**^{2}θ - [ 1 / cos^{2}θ ] + 1 = 0This 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 + tan

^{2}θ = sec

^{2}θ, 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 = [ sin^{2}θ / cos^{2}θ ] - [ 1 / cos^{2}θ ] + 1

Adding the fractions,

= [ sin^{2}θ - 1 + cos^{2}θ] / [ cos^{2}θ ]

Applying sin^{2}θ + cos^{2}θ = 1,

= [1 - 1] / [ cos^{2}θ ]

= 0 / [ cos^{2}θ ]

Zero divided by any quantity is zero,

= 0... which is the RHS

Method 2: Converting to secant function

From the identity 1 + tan^{2}θ = sec^{2}θ, it follows that tan^{2}θ = sec^{2}θ - 1,

LHS = [sec^{2}θ - 1] - [ 1 / cos^{2}θ ] + 1

Applying sec θ = 1 / cos θ, it implies that sec^{2}θ = 1 / cos^{2}θ

= [ sec^{2}θ - 1 ] - sec^{2}θ + 1

Simplifying,

= 0... which is the RHS

Trigonometric identities used in this post:

- tan θ = sin θ / cos θ
- sin
^{2}θ + cos^{2}θ = 1 - 1 + tan
^{2}θ = sec^{2}θ - sec θ = 1/cos θ

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