Question

Simplify the expression to a single trigonometric function:

a) cos(x) * sin(x) + sin(x)
b) cos(2x) + sin(2x)
c) tan(x) * cot(x)
d) sec(x) * csc(x)

Answer 1

The simplifications yield sin(x)(cos(x) + 1) for part a), no further simplification for part b), 1 for part c), and 2csc(2x) for part d). These are the simplest forms using standard trigonometric identities.

Step-by-step explanation:

To simplify the expression to a single trigonometric function for each part:

• a) cos(x) × sin(x) + sin(x)
  
  Here you can factor out sin(x), which gives you sin(x)(cos(x) + 1). This is the simplest form of the expression in terms of standard trigonometric identities.

• b) cos(2x) + sin(2x)
  
  This expression does not simplify to a single trigonometric function using standard identities. It is already in one of the simplest forms.

• c) tan(x) × cot(x)
  
  The identity tan(x) × cot(x) = 1 applies here, as cot(x) is the reciprocal of tan(x).

• d) sec(x) × csc(x)
  
  Since sec(x) = 1/cos(x) and csc(x) = 1/sin(x), the product sec(x) × csc(x) = 1/(sin(x)cos(x)).

  We use the identity sin(2x) = 2sin(x)cos(x), to write it as 1/(1/2)sin(2x) = 2/sin(2x), or simply 2csc(2x).