Question

Simplify the expression as much as possible: tan^3(x) - tan(x)sec^2(x).

Answer 1

To simplify the expression tan^3(x) - tan(x)sec^2(x), you can rewrite it using trigonometric identities, expand and combine like terms, and factor out common factors to arrive at the simplified expression -tan(x).

Step-by-step explanation:

To simplify the expression tan^3(x) - tan(x)sec^2(x), we can use the identities:

• tan^3(x) = (tan(x))^3
• sec^2(x) = 1 + tan^2(x)

Using these identities, we can rewrite the expression as:

(tan(x))^3 - tan(x)(1 + tan^2(x))

Expanding further:

(tan(x))^3 - tan(x) - tan^3(x)

Combining like terms:

-tan^3(x) + (tan(x))^3 - tan(x)

-tan(x) is common in all the terms, so we can factor it out:

tan(x)[(tan(x))^2 - 1 - tan^2(x)]

Simplifying further:

tan(x)[-1]

So the final simplified expression is:

-tan(x)