Trigonometric identities - 11

tan^4(x) + tan^2(x) = sec^4(x) - sec^2(x)

The left hand side of the equation contains tan raised to the power of 4 and 2. tan^4(x) can be written as (tan^2(x))^2, so the left hand side can be written as:

(tan^2(x))^2 + tan^2(x)

From the identity 1 + tan^2(x) = sec^2(x), it follows that tan^2(x) = sec^2(x) - 1.

(sec^2(x) - 1)^2 + sec^2(x) - 1

Apply the algebraic identity (a - b)^2 = a^2 + b^2 - 2ab to (sec^2(x) - 1)^2

sec^4(x) + 1 - 2sec^2(x) + sec^2(x) - 1

Simplify by adding like terms.

sec^4(x) - sec^2(x)

... which is the right hand side of the equation

Trigonometric identities applied:

• 1 + tan^2(x) = sec^2(x)