Find the multiplicative inverse of sec 𝜃 tan squared 𝜃 one sec 𝜃.
For a two-by-two matrix A, which is equal to 𝑎, 𝑏, 𝑐, 𝑑, its inverse is given by the formula one over the determinant of 𝑎 multiplied by 𝑑 negative 𝑏 and negative 𝑐 𝑎, where the determinant of A is given by 𝑎𝑑 minus 𝑏𝑐.
Notice that this means that if the determinant of the matrix A is zero, the inverse does not exist since one over the determinant of A is one over zero, which is undefined. Let’s call our matrix A and substitute the values of A into our formula for the determinant.
𝑎 multiplied by 𝑑 is sec 𝜃 multiplied by sec 𝜃. And 𝑏 multiplied by 𝑐 is tan squared 𝜃 multiplied by one. We can simplify this to get that the determinant of our matrix is sec squared 𝜃 minus tan squared 𝜃. Remember though that we know tan squared 𝜃 plus one is equal to sec squared 𝜃.
We can, therefore, replace sec squared 𝜃 and our expression for the determinant with tan squared 𝜃 plus one. tan squared 𝜃 minus tan squared 𝜃 is zero. So the determinant of our matrix is simply one.
Now, we can substitute everything we know into the formula for the multiplicative inverse of the matrix. That gives us one over one all multiplied by sec 𝜃 negative tan squared 𝜃 negative one sec 𝜃.
Since one over one is one, the multiplicative inverse of A is sec 𝜃 negative tan squared 𝜃 negative one sec 𝜃.