$$ \begin{array}{ll} ext { (b) } \frac{1}{ an heta}=\frac{\cos heta}{\sin heta} & ext { (An important result to remember) } \ ext { (c) } \frac{1}{\sin ^{2} heta}-\frac{\cos ^{2} heta}{\sin ^{2} heta}=1 & ext { (d) } \frac{ an x \cdot \cos x}{\sin x}=1\end{array} $$

Question

$$ \begin{array}{ll}	ext { (b) } \frac{1}{	an 	heta}=\frac{\cos 	heta}{\sin 	heta} & 	ext { (An important result to remember) } \ 	ext { (c) } \frac{1}{\sin ^{2} 	heta}-\frac{\cos ^{2} 	heta}{\sin ^{2} 	heta}=1 & 	ext { (d) } \frac{	an x \cdot \cos x}{\sin x}=1\end{array} $$

UpStudy AI · Accepted Answer

Let's analyze and simplify each of the given equations step by step.

### (b) 
The equation is:
$$
\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}
$$

**Step 1:** Recall the definition of tangent:
$$
\tan \theta = \frac{\sin \theta}{\cos \theta}
$$

**Step 2:** Substitute this definition into the left side:
$$
\frac{1}{\tan \theta} = \frac{1}{\frac{\sin \theta}{\cos \theta}} = \frac{\cos \theta}{\sin \theta}
$$

**Conclusion:** The equation holds true, confirming that $$\frac{1}{\tan \theta} = \cot \theta$$ and $$\frac{\cos \theta}{\sin \theta} = \cot \theta$$.

### (c)
The equation is:
$$
\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} = 1
$$

**Step 1:** Combine the fractions on the left side:
$$
\frac{1 - \cos^2 \theta}{\sin^2 \theta} = 1
$$

**Step 2:** Use the Pythagorean identity $$1 - \cos^2 \theta = \sin^2 \theta$$:
$$
\frac{\sin^2 \theta}{\sin^2 \theta} = 1
$$

**Conclusion:** This simplifies to $$1 = 1$$, confirming that the equation is valid.

### (d)
The equation is:
$$
\frac{\tan x \cdot \cos x}{\sin x} = 1
$$

**Step 1:** Recall the definition of tangent:
$$
\tan x = \frac{\sin x}{\cos x}
$$

**Step 2:** Substitute this definition into the left side:
$$
\frac{\frac{\sin x}{\cos x} \cdot \cos x}{\sin x} = \frac{\sin x}{\sin x} = 1
$$

**Conclusion:** This simplifies to $$1 = 1$$, confirming that the equation is valid.

### Summary
All three equations are valid and demonstrate important trigonometric identities:
- (b) confirms the relationship between cotangent and tangent.
- (c) illustrates a Pythagorean identity.
- (d) shows the relationship between tangent and sine/cosine.
All three equations are valid trigonometric identities: - (b) $$\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$$ - (c) $$\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} = 1$$ - (d) $$\frac{\tan x \cdot \cos x}{\sin x} = 1$$

\( \begin{array}{ll}\text { (b) } \frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \text { (An important result to remember) } \\ \text { (c) } \frac{1}{\sin ^{2} \theta}-\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=1 & \text { (d) } \frac{\tan x \cdot \cos x}{\sin x}=1\end{array} \)

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