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:

• tan θ = sin θ / cos θ
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• sec θ = 1/cos θ