$$ 1 \frac{\sin x}{\cos x \cdot an x} $$ 2) $$ an ^{2} x\left(1-\sin ^{2} x ight) $$ \% $$ \frac{\left(1-\cos ^{2} x ight)}{\sin x} $$ $$ 48 \sin ^{2} heta+8 \cos ^{2} heta $$ 3 $$ 1- an x \cdot \cos x \cdot \sin x $$ $$ 7 \frac{1}{\sin x}-\frac{\cos x}{ an x} $$ $$ 6 \quad(3-3 \sin heta)(3+3 \sin heta) $$ $$ 8\left(\frac{1}{ an x}+ an x ight)(\sin x \cdot \cos x) $$ 9) $$ \frac{\sin x \cdot \cos x}{1+\cos ^{2} x-\sin ^{2} x} $$ $$ 10 an ^{2} x-\frac{1}{\cos ^{2} x} $$

Question

$$ 1 \frac{\sin x}{\cos x \cdot 	an x} $$ 2) $$ 	an ^{2} x\left(1-\sin ^{2} xight) $$ \% $$ \frac{\left(1-\cos ^{2} xight)}{\sin x} $$ $$ 48 \sin ^{2} 	heta+8 \cos ^{2} 	heta $$ 3 $$ 1-	an x \cdot \cos x \cdot \sin x $$ $$ 7 \frac{1}{\sin x}-\frac{\cos x}{	an x} $$ $$ 6 \quad(3-3 \sin 	heta)(3+3 \sin 	heta) $$ $$ 8\left(\frac{1}{	an x}+	an xight)(\sin x \cdot \cos x) $$ 9) $$ \frac{\sin x \cdot \cos x}{1+\cos ^{2} x-\sin ^{2} x} $$ $$ 10 	an ^{2} x-\frac{1}{\cos ^{2} x} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$1-	an\left(xight)\cos\left(xight)\sin\left(xight)$$
- step1: Transform the expression:
  $$1-\sin^{2}\left(xight)$$
- step2: Transform the expression:
  $$\cos^{2}\left(xight)$$
Calculate or simplify the expression $$ 	an(x)^2 - (1/\cos(x)^2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$	an\left(x^{2}ight)-\left(\frac{1}{\cos\left(x^{2}ight)}ight)$$
- step1: Remove the parentheses:
  $$	an\left(x^{2}ight)-\frac{1}{\cos\left(x^{2}ight)}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{	an\left(x^{2}ight)\cos\left(x^{2}ight)}{\cos\left(x^{2}ight)}-\frac{1}{\cos\left(x^{2}ight)}$$
- step3: Transform the expression:
  $$\frac{	an\left(x^{2}ight)\cos\left(x^{2}ight)-1}{\cos\left(x^{2}ight)}$$
Calculate or simplify the expression $$ 1 * (\sin(x) / (\cos(x) * 	an(x))) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$1	imes \left(\frac{\sin\left(xight)}{\left(\cos\left(xight)	an\left(xight)ight)}ight)$$
- step1: Remove the parentheses:
  $$1	imes \left(\frac{\sin\left(xight)}{\cos\left(xight)	an\left(xight)}ight)$$
- step2: Remove the parentheses:
  $$1	imes \frac{\sin\left(xight)}{\cos\left(xight)	an\left(xight)}$$
- step3: Calculate:
  $$\frac{\sin\left(xight)}{\cos\left(xight)	an\left(xight)}$$
- step4: Transform the expression:
  $$\frac{\sin\left(xight)}{\sin\left(xight)}$$
- step5: Reduce the fraction:
  $$1$$
Calculate or simplify the expression $$ 6 * (3 - 3 * \sin(	heta)) * (3 + 3 * \sin(	heta)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$6\left(3-3\sin\left(	heta ight)ight)\left(3+3\sin\left(	heta ight)ight)$$
- step1: Multiply the first two terms:
  $$\left(18-18\sin\left(	heta ight)ight)\left(3+3\sin\left(	heta ight)ight)$$
- step2: Transform the expression:
  $$54-54\sin^{2}\left(	heta ight)$$
- step3: Transform the expression:
  $$54-54+54\cos^{2}\left(	heta ight)$$
- step4: Remove the opposites:
  $$54\cos^{2}\left(	heta ight)$$
Calculate or simplify the expression $$ 	an(x)^2 * (1 - \sin(x)^2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$	an\left(x^{2}ight)\left(1-\sin\left(x^{2}ight)ight)$$
- step1: Calculate:
  $$\frac{\sin\left(x^{2}ight)}{\cos\left(x^{2}ight)}	imes \left(1-\sin\left(x^{2}ight)ight)$$
- step2: Calculate:
  $$\frac{\sin\left(x^{2}ight)\left(1-\sin\left(x^{2}ight)ight)}{\cos\left(x^{2}ight)}$$
- step3: Rewrite the expression:
  $$\left(1-\sin\left(x^{2}ight)ight)\cos^{-1}\left(x^{2}ight)\sin\left(x^{2}ight)$$
- step4: Rewrite the expression:
  $$\sin\left(x^{2}ight)\cos^{-1}\left(x^{2}ight)-\cos^{-1}\left(x^{2}ight)\sin^{2}\left(x^{2}ight)$$
- step5: Rewrite the expression:
  $$\cos^{-1}\left(x^{2}ight)\sin\left(x^{2}ight)-\sin^{2}\left(x^{2}ight)\cos^{-1}\left(x^{2}ight)$$
- step6: Simplify:
  $$	an\left(x^{2}ight)-\sin^{2}\left(x^{2}ight)\cos^{-1}\left(x^{2}ight)$$
- step7: Rewrite the expression:
  $$	an\left(x^{2}ight)-\cos^{-1}\left(x^{2}ight)\sin^{2}\left(x^{2}ight)$$
- step8: Simplify:
  $$	an\left(x^{2}ight)-\sec\left(x^{2}ight)\sin^{2}\left(x^{2}ight)$$
- step9: Rewrite the expression:
  $$	an\left(x^{2}ight)-\sin^{2}\left(x^{2}ight)\sec\left(x^{2}ight)$$
Calculate or simplify the expression $$ (1 - \cos(x)^2) / \sin(x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(1-\cos\left(x^{2}ight)ight)}{\sin\left(xight)}$$
- step1: Remove the parentheses:
  $$\frac{1-\cos\left(x^{2}ight)}{\sin\left(xight)}$$
- step2: Rewrite the expression:
  $$\left(1-\cos\left(x^{2}ight)ight)\sin^{-1}\left(xight)$$
- step3: Rewrite the expression:
  $$\sin^{-1}\left(xight)-\sin^{-1}\left(xight)\cos\left(x^{2}ight)$$
- step4: Calculate:
  $$\csc\left(xight)-\sin^{-1}\left(xight)\cos\left(x^{2}ight)$$
- step5: Calculate:
  $$\csc\left(xight)-\csc\left(xight)\cos\left(x^{2}ight)$$
- step6: Rewrite the expression:
  $$\csc\left(xight)-\cos\left(x^{2}ight)\csc\left(xight)$$
Calculate or simplify the expression $$ 7 * (1/\sin(x)) - (\cos(x)/	an(x)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$7\left(\frac{1}{\sin\left(xight)}ight)-\left(\frac{\cos\left(xight)}{	an\left(xight)}ight)$$
- step1: Remove the parentheses:
  $$7	imes \frac{1}{\sin\left(xight)}-\left(\frac{\cos\left(xight)}{	an\left(xight)}ight)$$
- step2: Remove the parentheses:
  $$7	imes \frac{1}{\sin\left(xight)}-\frac{\cos\left(xight)}{	an\left(xight)}$$
- step3: Multiply the terms:
  $$\frac{7}{\sin\left(xight)}-\frac{\cos\left(xight)}{	an\left(xight)}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{7	an\left(xight)}{\sin\left(xight)	an\left(xight)}-\frac{\cos\left(xight)\sin\left(xight)}{	an\left(xight)\sin\left(xight)}$$
- step5: Rewrite the expression:
  $$\frac{7	an\left(xight)}{\sin\left(xight)	an\left(xight)}-\frac{\cos\left(xight)\sin\left(xight)}{\sin\left(xight)	an\left(xight)}$$
- step6: Transform the expression:
  $$\frac{7	an\left(xight)-\cos\left(xight)\sin\left(xight)}{\sin\left(xight)	an\left(xight)}$$
- step7: Transform the expression:
  $$\frac{\frac{7\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}}{\sin\left(xight)	an\left(xight)}$$
- step8: Transform the expression:
  $$\frac{\frac{7\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}}{\frac{\sin^{2}\left(xight)}{\cos\left(xight)}}$$
- step9: Multiply by the reciprocal:
  $$\frac{7\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}	imes \frac{\cos\left(xight)}{\sin^{2}\left(xight)}$$
- step10: Rewrite the expression:
  $$\frac{\sin\left(xight)\left(7-\cos^{2}\left(xight)ight)}{\cos\left(xight)}	imes \frac{\cos\left(xight)}{\sin^{2}\left(xight)}$$
- step11: Reduce the fraction:
  $$\left(7-\cos^{2}\left(xight)ight)	imes \frac{1}{\sin\left(xight)}$$
- step12: Multiply the terms:
  $$\frac{7-\cos^{2}\left(xight)}{\sin\left(xight)}$$
- step13: Transform the expression:
  $$\frac{6+\sin^{2}\left(xight)}{\sin\left(xight)}$$
- step14: Transform the expression:
  $$\frac{6}{\sin\left(xight)}+\sin\left(xight)$$
- step15: Transform the expression:
  $$6\csc\left(xight)+\sin\left(xight)$$
Calculate or simplify the expression $$ 48 * \sin(	heta)^2 + 8 * \cos(	heta)^2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$48\sin\left(	heta ^{2}ight)+8\cos\left(	heta ^{2}ight)$$
Calculate or simplify the expression $$ 8 * ((1/	an(x)) + 	an(x)) * (\sin(x) * \cos(x)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$8\left(\left(\frac{1}{	an\left(xight)}ight)+	an\left(xight)ight)\left(\sin\left(xight)\cos\left(xight)ight)$$
- step1: Remove the parentheses:
  $$8\left(\left(\frac{1}{	an\left(xight)}ight)+	an\left(xight)ight)\sin\left(xight)\cos\left(xight)$$
- step2: Remove the parentheses:
  $$8\left(\frac{1}{	an\left(xight)}+	an\left(xight)ight)\sin\left(xight)\cos\left(xight)$$
- step3: Add the terms:
  $$8	imes \frac{1+	an^{2}\left(xight)}{	an\left(xight)}	imes \sin\left(xight)\cos\left(xight)$$
- step4: Multiply the terms:
  $$\frac{8+8	an^{2}\left(xight)}{	an\left(xight)}	imes \sin\left(xight)\cos\left(xight)$$
- step5: Multiply the terms:
  $$\frac{\left(8+8	an^{2}\left(xight)ight)\sin\left(xight)\cos\left(xight)}{	an\left(xight)}$$
- step6: Transform the expression:
  $$\frac{\left(8+8\left(\frac{\sin\left(xight)}{\cos\left(xight)}ight)^{2}ight)\sin\left(xight)\cos\left(xight)}{	an\left(xight)}$$
- step7: Transform the expression:
  $$\frac{\left(8+8\left(\frac{\sin\left(xight)}{\cos\left(xight)}ight)^{2}ight)\sin\left(xight)\cos\left(xight)}{\frac{\sin\left(xight)}{\cos\left(xight)}}$$
- step8: Rewrite the expression:
  $$\frac{\frac{\left(8\cos^{2}\left(xight)+8\sin^{2}\left(xight)ight)\sin\left(xight)\cos\left(xight)}{\cos^{2}\left(xight)}}{\frac{\sin\left(xight)}{\cos\left(xight)}}$$
- step9: Multiply by the reciprocal:
  $$\frac{\left(8\cos^{2}\left(xight)+8\sin^{2}\left(xight)ight)\sin\left(xight)\cos\left(xight)}{\cos^{2}\left(xight)}	imes \frac{\cos\left(xight)}{\sin\left(xight)}$$
- step10: Reduce the fraction:
  $$\frac{\left(8\cos^{2}\left(xight)+8\sin^{2}\left(xight)ight)\cos\left(xight)}{\cos\left(xight)}	imes 1$$
- step11: Multiply the terms:
  $$\frac{\left(8\cos^{2}\left(xight)+8\sin^{2}\left(xight)ight)\cos\left(xight)}{\cos\left(xight)}$$
- step12: Reduce the fraction:
  $$8\cos^{2}\left(xight)+8\sin^{2}\left(xight)$$
- step13: Transform the expression:
  $$8\cos^{2}\left(xight)+8-8\cos^{2}\left(xight)$$
- step14: Collect like terms:
  $$8$$
Calculate or simplify the expression $$ (\sin(x) * \cos(x)) / (1 + \cos(x)^2 - \sin(x)^2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\sin\left(xight)\cos\left(xight)ight)}{\left(1+\cos\left(x^{2}ight)-\sin\left(x^{2}ight)ight)}$$
- step1: Remove the parentheses:
  $$\frac{\sin\left(xight)\cos\left(xight)}{1+\cos\left(x^{2}ight)-\sin\left(x^{2}ight)}$$
- step2: Transform the expression:
  $$\frac{\frac{\sin\left(2xight)}{2}}{1+\cos\left(x^{2}ight)-\sin\left(x^{2}ight)}$$
- step3: Multiply by the reciprocal:
  $$\frac{\sin\left(2xight)}{2}	imes \frac{1}{1+\cos\left(x^{2}ight)-\sin\left(x^{2}ight)}$$
- step4: Multiply the terms:
  $$\frac{\sin\left(2xight)}{2\left(1+\cos\left(x^{2}ight)-\sin\left(x^{2}ight)ight)}$$
- step5: Multiply the terms:
  $$\frac{\sin\left(2xight)}{2+2\cos\left(x^{2}ight)-2\sin\left(x^{2}ight)}$$
Here are the simplified results for each expression you provided:

1. **Expression:** $$ 1 \frac{\sin x}{\cos x \cdot 	an x} $$  
   **Simplified Result:** $$ 1 $$

2. **Expression:** $$ 	an ^{2} x\left(1-\sin ^{2} xight) $$  
   **Simplified Result:** $$ 	an(x^{2}) - \sin^{2}(x^{2}) \sec(x^{2}) $$

3. **Expression:** $$ \frac{1-\cos ^{2} x}{\sin x} $$  
   **Simplified Result:** $$ \csc(x) - \cos(x^{2}) \csc(x) $$

4. **Expression:** $$ 48 \sin ^{2} 	heta + 8 \cos ^{2} 	heta $$  
   **Simplified Result:** $$ 48 \sin(	heta^{2}) + 8 \cos(	heta^{2}) $$

5. **Expression:** $$ 1 - 	an x \cdot \cos x \cdot \sin x $$  
   **Simplified Result:** $$ \cos^{2}(x) $$

6. **Expression:** $$ 7 \frac{1}{\sin x} - \frac{\cos x}{	an x} $$  
   **Simplified Result:** $$ 6 \csc(x) + \sin(x) $$

7. **Expression:** $$ 6 \quad(3-3 \sin 	heta)(3+3 \sin 	heta) $$  
   **Simplified Result:** $$ 54 \cos^{2}(	heta) $$

8. **Expression:** $$ 8\left(\frac{1}{	an x}+	an xight)(\sin x \cdot \cos x) $$  
   **Simplified Result:** $$ 8 $$

9. **Expression:** $$ \frac{\sin x \cdot \cos x}{1+\cos ^{2} x-\sin ^{2} x} $$  
   **Simplified Result:** $$ \frac{\sin(2x)}{2 + 2\cos(x^{2}) - 2\sin(x^{2})} $$

10. **Expression:** $$ 10 	an ^{2} x - \frac{1}{\cos ^{2} x} $$  
    **Simplified Result:** $$ \frac{	an(x^{2}) \cos(x^{2}) - 1}{\cos(x^{2})} $$

If you have any further questions or need additional assistance, feel free to ask!
Here are the simplified results for each expression: 1. $$ 1 \frac{\sin x}{\cos x \cdot 	an x} = 1 $$ 2. $$ 	an ^{2} x\left(1-\sin ^{2} xight) = 	an(x^{2}) - \sin^{2}(x^{2}) \sec(x^{2}) $$ 3. $$ \frac{1-\cos ^{2} x}{\sin x} = \csc(x) - \cos(x^{2}) \csc(x) $$ 4. $$ 48 \sin ^{2} 	heta + 8 \cos ^{2} 	heta = 48 \sin(	heta^{2}) + 8 \cos(	heta^{2}) $$ 5. $$ 1 - 	an x \cdot \cos x \cdot \sin x = \cos^{2}(x) $$ 6. $$ 7 \frac{1}{\sin x} - \frac{\cos x}{	an x} = 6 \csc(x) + \sin(x) $$ 7. $$ 6 \quad(3-3 \sin 	heta)(3+3 \sin 	heta) = 54 \cos^{2}(	heta) $$ 8. $$ 8\left(\frac{1}{	an x}+	an xight)(\sin x \cdot \cos x) = 8 $$ 9. $$ \frac{\sin x \cdot \cos x}{1+\cos ^{2} x-\sin ^{2} x} = \frac{\sin(2x)}{2 + 2\cos(x^{2}) - 2\sin(x^{2})} $$ 10. $$ 10 	an ^{2} x - \frac{1}{\cos ^{2} x} = \frac{	an(x^{2}) \cos(x^{2}) - 1}{\cos(x^{2})} $$

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