Set up the determinant by breaking it into smaller components.

Since the matrix is multiplied by , the determinant is .

Since the matrix is multiplied by , the determinant is .

The determinant of a matrix can be found using the formula .

Simplify the determinant.

Rewrite in terms of sines and cosines.

Simplify each term.

Multiply by .

Rewrite as .

Simplify terms.

Apply the distributive property.

Cancel the common factor of .

Move the leading negative in into the numerator.

Cancel the common factor.

Rewrite the expression.

Multiply .

Multiply by .

Multiply by .

Combine and .

Convert from to .

Add and .

Subtract from .

Find the Determinant [[1,0,sin(x)],[1,0,cos(x)],[1,sec(x),-sin(x)]]