Question

Can someone please explain this problem step by step? Thank you sooooooooooooo much!

Answer 1

Explanation:

Use the Pythagorean identity, cancel common factors, and divide numerator and denominator by cos(x). Equivalently, multiply numerator and denominator by sec(x).

$\dfrac{\sin^2{x}+2\cos{x}-1}{\sin^2{x}+3\cos{x}-3}=\dfrac{1-\cos^2{x}+2\cos{x}-1}{1-\cos^2{x}+3\cos{x}-3} \qquad\text{replace \sin^2 with 1-\cos^2 }\\\\=\dfrac{\cos{x}(2-\cos{x})}{(\cos{x}-1)(2-\cos{x})}=\dfrac{\cos{x}}{\cos{x}-1}=\dfrac{1}{\dfrac{\cos{x}}{\cos{x}}-\dfrac{1}{\cos{x}}}=\dfrac{1}{1-\sec{x}}$