Math Snap

STEP 1

1. We are working with trigonometric identities and relationships.
2. We will use basic trigonometric identities to express one function in terms of others.
3. We assume $x$ is an angle where all trigonometric functions are defined.

STEP 2

1. Express $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$.
2. Express $\csc(x)$ in terms of $\cos(x)$ and $\tan(x)$.

STEP 3

Start with the identity for $\cot(x)$ and $\csc(x)$:
$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$ $$\csc(x) = \frac{1}{\sin(x)}$$

STEP 4

Solve the expression for $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$:
Since $\cot(x) = \frac{\cos(x)}{\sin(x)}$, we can express $\sin(x)$ in terms of $\csc(x)$:
$$\sin(x) = \frac{1}{\csc(x)}$$ Substitute $\sin(x)$ into the expression for $\cot(x)$:
$$\cot(x) = \frac{\cos(x)}{\frac{1}{\csc(x)}} = \cos(x) \cdot \csc(x)$$ Thus, $\cos(x) = \cot(x) \cdot \frac{1}{\csc(x)} = \frac{\cot(x)}{\csc(x)}$.

STEP 5

Start with the identity for $\tan(x)$ and $\cos(x)$:
$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

Express $\sin(x)$ in terms of $\cos(x)$ and $\tan(x)$:
Rearrange the expression for $\tan(x)$:
$$\sin(x) = \tan(x) \cdot \cos(x)$$ Since $\csc(x) = \frac{1}{\sin(x)}$, substitute the expression for $\sin(x)$:
$$\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$$ Thus, $\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$.
The expressions are:
a. $\cos(x) = \frac{\cot(x)}{\csc(x)}$
b. $\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$