Help answer this question below.

Question

Simplify tanx - sin^3xsecx

Answer 1

tan(x)-sin(x)^3*sec(x)

= [sin(x)/cos(x)]-[{sin(x)^3}/cos(x)]    [Since tan(x) = sin(x)/cos(x), sec(x) = 1/cos(x)]

= [(sin(x) - sin(x)^3] / cos(x)                 

= [sin(x) (1 - sin(x)^2)] / cos(x)           [Tanking sin(x) as common]

= [sin(x)*cos(x)^2]/cos(x)                    [Using the trig identity 1-sin(x)^2=cos(x)^2]

= sin(x) cos(x)        [Divide out the common cos(x) factor in the numerator, denominator]

= (2/2)sin(x)cos(x)                             [Multiply and divide by 2]

= 1/2*sin(2x)                                      [Since sin(2x)=2sin(x)cos(x)]

Therefore tan(x)-sin(x)^3*sec(x) = 1/2*sin(2x)