Prove the identities. $$ 1 \quad \sin x \cdot \cos x \cdot an x=1-\cos ^{2} x $$ $$ 2 \quad \cos ^{3} x+\cos x \cdot \sin ^{2} x=\cos x $$ $$ 3 \quad(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=2 $$ $$ 4 \quad\left(1-\sin ^{2} x ight)\left(1+ an ^{2} x ight)=1 $$ $$ 5 \quad \frac{1}{\sin x}-\sin x=\frac{\cos x}{ an x} $$ $$ 6 \quad \frac{\cos x-\cos ^{3} x}{\cos x-\cos x \cdot \sin ^{2} x}= an ^{2} x $$ $$ 7 \quad \frac{\sin heta}{ an heta}-\sqrt{1-\sin ^{2} heta}=0 $$ $$ 8 \quad an ^{2} heta+1=\frac{1}{\cos ^{2} heta} $$

Question

Prove the identities. $$ 1 \quad \sin x \cdot \cos x \cdot 	an x=1-\cos ^{2} x $$ $$ 2 \quad \cos ^{3} x+\cos x \cdot \sin ^{2} x=\cos x $$ $$ 3 \quad(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=2 $$ $$ 4 \quad\left(1-\sin ^{2} xight)\left(1+	an ^{2} xight)=1 $$ $$ 5 \quad \frac{1}{\sin x}-\sin x=\frac{\cos x}{	an x} $$ $$ 6 \quad \frac{\cos x-\cos ^{3} x}{\cos x-\cos x \cdot \sin ^{2} x}=	an ^{2} x $$ $$ 7 \quad \frac{\sin 	heta}{	an 	heta}-\sqrt{1-\sin ^{2} 	heta}=0 $$ $$ 8 \quad 	an ^{2} 	heta+1=\frac{1}{\cos ^{2} 	heta} $$

UpStudy AI · Accepted Answer

Verify the identity by following steps:
- step0: Verify:
  $$\cos^{3}\left(xight)+\cos\left(xight)\sin^{2}\left(xight)=\cos\left(xight)$$
- step1: Choose a side to work on:
  $$\cos\left(xight)=\cos\left(xight)$$
- step2: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ 	an^2(	heta) + 1 = \frac{1}{\cos^2(	heta)} $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$	an^{2}\left(	heta ight)+1=\frac{1}{\cos^{2}\left(	heta ight)}$$
- step1: Choose a side to work on:
  $$1+	an^{2}\left(	heta ight)=\frac{1}{\cos^{2}\left(	heta ight)}$$
- step2: Choose the other side to work on:
  $$1+	an^{2}\left(	heta ight)=1+	an^{2}\left(	heta ight)$$
- step3: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ (1 - \sin^2(x))(1 + 	an^2(x)) = 1 $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\left(1-\sin^{2}\left(xight)ight)\left(1+	an^{2}\left(xight)ight)=1$$
- step1: Choose a side to work on:
  $$1=1$$
- step2: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ \sin(x) \cos(x) 	an(x) = 1 - \cos^2(x) $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\sin\left(xight)\cos\left(xight)	an\left(xight)=1-\cos^{2}\left(xight)$$
- step1: Choose a side to work on:
  $$\sin^{2}\left(xight)=1-\cos^{2}\left(xight)$$
- step2: Choose the other side to work on:
  $$\sin^{2}\left(xight)=\sin^{2}\left(xight)$$
- step3: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ \frac{1}{\sin(x)} - \sin(x) = \frac{\cos(x)}{	an(x) } $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\frac{1}{\sin\left(xight)}-\sin\left(xight)=\frac{\cos\left(xight)}{	an\left(xight)}$$
- step1: Choose a side to work on:
  $$\frac{\cos^{2}\left(xight)}{\sin\left(xight)}=\frac{\cos\left(xight)}{	an\left(xight)}$$
- step2: Choose the other side to work on:
  $$\frac{\cos^{2}\left(xight)}{\sin\left(xight)}=\frac{\cos^{2}\left(xight)}{\sin\left(xight)}$$
- step3: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ (\sin(x) + \cos(x))^2 + (\sin(x) - \cos(x))^2 = 2 $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\left(\sin\left(xight)+\cos\left(xight)ight)^{2}+\left(\sin\left(xight)-\cos\left(xight)ight)^{2}=2$$
- step1: Choose a side to work on:
  $$2=2$$
- step2: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ \frac{\cos(x) - \cos^3(x)}{\cos(x) - \cos(x) \sin^2(x)} = 	an^2(x) $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\frac{\cos\left(xight)-\cos^{3}\left(xight)}{\cos\left(xight)-\cos\left(xight)\sin^{2}\left(xight)}=	an^{2}\left(xight)$$
- step1: Choose a side to work on:
  $$	an^{2}\left(xight)=	an^{2}\left(xight)$$
- step2: Verify the identity:
  $$	extrm{true}$$
Determine whether the expression $$ \frac{\sin(	heta)}{	an(	heta)} - \sqrt(1 - \sin^2(	heta)) = 0 $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\frac{\sin\left(	heta ight)}{	an\left(	heta ight)}-\sqrt{1-\sin^{2}\left(	heta ight)}=0$$
- step1: Choose a side to work on:
  $$\cos\left(	heta ight)-\left|\cos\left(	heta ight)ight|=0$$
- step2: Verify the identity:
  $$	extrm{false}$$
Here are the results for the identities you provided:

1. **Identity:** $$ \sin x \cdot \cos x \cdot 	an x = 1 - \cos^2 x $$  
   **Result:** True

2. **Identity:** $$ \cos^3 x + \cos x \cdot \sin^2 x = \cos x $$  
   **Result:** True

3. **Identity:** $$ (\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2 $$  
   **Result:** True

4. **Identity:** $$ (1 - \sin^2 x)(1 + 	an^2 x) = 1 $$  
   **Result:** True

5. **Identity:** $$ \frac{1}{\sin x} - \sin x = \frac{\cos x}{	an x} $$  
   **Result:** True

6. **Identity:** $$ \frac{\cos x - \cos^3 x}{\cos x - \cos x \cdot \sin^2 x} = 	an^2 x $$  
   **Result:** True

7. **Identity:** $$ \frac{\sin 	heta}{	an 	heta} - \sqrt{1 - \sin^2 	heta} = 0 $$  
   **Result:** False

8. **Identity:** $$ 	an^2 	heta + 1 = \frac{1}{\cos^2 	heta} $$  
   **Result:** True

### Summary:
- Identities 1 to 6, and 8 are true.
- Identity 7 is false.
Identities 1, 2, 3, 4, 5, 6, and 8 are true. Identity 7 is false.

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