23 Trigonometric identities and functions; Question 23.19

[Eigenvalue]

23.19 Simplify:
a) sin²(x)cot²(x)

b)cos(x)(1+tan²(x))

c) (cot(x)+csc(x))(cot(x)-csc(x))

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

[Eigenvalue]

a) sin²(x)cot²(x)
=sin²(x)[tex]\frac{cos²(x)}{sin²(x)}[/tex]
=cos²(x)

b)cos(x)(1+tan²(x))
=cos(x)sec²(x)
=[tex]\frac{cos(x)}{cos²(x)}[/tex]

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

=sec(x)

c) (cot(x)+csc(x))(cot(x)-csc(x))
=cot²(x)-csc²(x)
=-(csc²(x)-cot²(x))
=-1

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

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

=[tex]\frac{2sin(x)cos(x)}{1-sin²(x)}[/tex]

=[tex]\frac{2sin(x)cos(x)}{cos²(x)}[/tex]

=[tex]\frac{2sin(x)}{cos(x)}[/tex]

=2tan(x)