How do you factor sec^2(x)-sec(x)+sin^2(x)*sec^2(x)?

Question

James Adkins · Accepted Answer

#sec^2x-secx+sin^2xsec^2x=-sec^2x(cosx+2)(cosx-1)# Start by factoring out #-sec^2x# to get #-sec^2x(-1+1/secx-sin^2x)# Using the fact that #secx=1/cosx# and #sin^2x=1-cos^2x#, we get #-sec^2x(-1+cosx-1+cos^2x)# Rearranging gives #-sec^2x(cos^2x-cosx-2)# Recognizing that this is a quadratic expression of #cosx#, we can factor to get #-sec^2x(cosx+2)(cosx-1)#

Daniel Acosta · Answer

undefined To factor the expression $ \sec^2(x) - \sec(x) + \sin^2(x) \cdot \sec^2(x) $, you can rewrite it as $ \sec^2(x) \left(1 - \frac{1}{\sec(x)} + \sin^2(x)\right) $. Then, using the trigonometric identity $ \sin^2(x) = 1 - \cos^2(x) $, the expression becomes $ \sec^2(x) \left(1 - \frac{1}{\sec(x)} + 1 - \cos^2(x)\right) $. Simplifying further, it becomes $ \sec^2(x) \left(2 - \frac{1}{\sec(x)} - \cos^2(x)\right) $. Finally, using the Pythagorean identity $ \cos^2(x) = 1 - \sin^2(x) $, the expression factors into $ \sec^2(x) \left(2 - \frac{1}{\sec(x)} - (1 - \sin^2(x))\right) $. This further simplifies to $ \sec^2(x) \left(2 - \frac{1}{\sec(x)} - (1 - (1 - \cos^2(x)))\right) $. And finally, it simplifies to $ \sec^2(x) \left(2 - \frac{1}{\sec(x)} - (1 - (1 - (1 - \sin^2(x))))\right) $.