Question

math 30 pre - calculus
chapter 6 review
name:
answer all questions in the space provided. you must show a full solution in order to receive full marks. good luck!

1. rewrite each trigonometric expression in terms of sine or cosine or both. then, simplify as much as possible.

a) (\tan x cot x)
b) (\frac{1}{csc^{2} x} + \frac{1}{sec^{2} x})

Explanation:

Part a)

Step1: Recall trigonometric identities

We know that \(\tan x=\frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\).

Step2: Substitute the identities into the expression

Substitute these into \(\tan x\cot x\):

$$ \tan x\cot x=\frac{\sin x}{\cos x}\times\frac{\cos x}{\sin x} $$

Step3: Simplify the expression

The \(\sin x\) in the numerator and denominator cancels out, and the \(\cos x\) in the numerator and denominator also cancels out, so we get \(1\).

Step1: Recall reciprocal identities

We know that \(\csc x=\frac{1}{\sin x}\), so \(\csc^{2}x = \frac{1}{\sin^{2}x}\), and \(\sec x=\frac{1}{\cos x}\), so \(\sec^{2}x=\frac{1}{\cos^{2}x}\).

Step2: Substitute the identities into the expression

Substitute these into \(\frac{1}{\csc^{2}x}+\frac{1}{\sec^{2}x}\):

$$ \frac{1}{\csc^{2}x}+\frac{1}{\sec^{2}x}=\sin^{2}x+\cos^{2}x $$

Step3: Use the Pythagorean identity

We know from the Pythagorean identity that \(\sin^{2}x+\cos^{2}x = 1\).

Answer:

\(1\)

Part b)