23 Trigonometric identities and functions; Question 23.20

[Eigenvalue]

23.20. Simplify

a)csc(x)tan(x)

b)1-[tex]\frac{sin²(x)}{1+cos(x)}[/tex]

c)[tex]\frac{sin⁴(x)-cos⁴(x)}{sin(x)+cos(x)}[/tex]

d) [tex]\frac{sec(x)}{csc(x)}[/tex]+[tex]\frac{sin(x)}{cos(x)}[/tex]

[Eigenvalue]

a)csc(x)tan(x)
=[tex]\frac{1}{sin(x)}[/tex]*[tex]\frac{sin(x)}{cos(x)}[/tex]

=[tex]\frac{1}{cos(x)}[/tex]

=sec(x)

b)1-[tex]\frac{sin²(x)}{1+cos(x)}[/tex]

=[tex]\frac{1+cos(x)-sin²(x)}{1+cos(x)}[/tex]

=[tex]\frac{cos²(x)+cos(x)}{1+cos(x)}[/tex]

=cos(x)

c)[tex]\frac{sin⁴(x)-cos⁴(x)}{sin(x)+cos(x)}[/tex]
=(sin²(x)+cos²(x))(sin(x)-cos(x))
=sin(x)-cos(x)

d) [tex]\frac{sec(x)}{csc(x)}[/tex]+[tex]\frac{sin(x)}{cos(x)}[/tex]

=[tex]\frac{1/cos(x)}{1/sin(x)}[/tex]+[tex]\frac{sin(x)}{cos(x)}[/tex]

=[tex]\frac{sin(x)}{cos(x)}[/tex]+[tex]\frac{sin(x)}{cos(x)}[/tex]

=2tan(x)