4sinx-5cosx+3sec²x

Answer 1

To simplify the expression 4sin(x) - 5cos(x) + 3sec^2(x), we can start by using the identities:

sin(x) = 1 / csc(x)
cos(x) = 1 / sec(x)

Substituting these identities into the expression, we get:

4(1 / csc(x)) - 5(1 / sec(x)) + 3sec^2(x)

= 4 / csc(x) - 5 / sec(x) + 3sec^2(x)

To further simplify, we can convert the csc(x) and sec(x) terms to sin(x) and cos(x) respectively.

Recall that csc(x) = 1 / sin(x) and sec(x) = 1 / cos(x).

Substituting these identities, we get:

4sin(x) - 5cos(x) + 3(1 / cos^2(x))

= 4sin(x) - 5cos(x) + 3 / cos^2(x)

= 4sin(x) - 5cos(x) + 3sec^2(x)

So the simplified form of the expression 4sin(x) - 5cos(x) + 3sec^2(x) is 4sin(x) - 5cos(x) + 3 / cos^2(x).