\( \left. \begin{array} { l } { \tan ^ { 2 } x ( 1 - \sin ^ { 2 } x ) } \\ { 8 \sin ^ { 2 } \theta + 8 \cos ^ { 2 } \theta } \\ { ( 3 - 3 \sin \theta ) ( 3 + 3 \sin \theta ) } \\ { ( \frac { 1 } { \tan x } + \tan x ) ( \sin x \cdot \cos x ) } \\ { \tan ^ { 2 } x - \frac { 1 } { \cos ^ { 2 } x } } \\ \hline \end{array} \right. \)

Simplify the expression by following steps: - step0: Solution: \(8\sin^{2}\left(\theta \right)+8\cos^{2}\left(\theta \right)\) - step1: Transform the expression: \(8-8\cos^{2}\left(\theta \right)+8\cos^{2}\left(\theta \right)\) - step2: Collect like terms: \(8\) Calculate or simplify the expression \( (3 - 3 * \sin(\theta)) * (3 + 3 * \sin(\theta)) \). Simplify the expression by following steps: - step0: Solution: \(\left(3-3\sin\left(\theta \right)\right)\left(3+3\sin\left(\theta \right)\right)\) - step1: Transform the expression: \(9-9\sin^{2}\left(\theta \right)\) - step2: Transform the expression: \(9-9+9\cos^{2}\left(\theta \right)\) - step3: Remove the opposites: \(9\cos^{2}\left(\theta \right)\) Calculate or simplify the expression \( \tan^2(x) - 1/\cos^2(x) \). Simplify the expression by following steps: - step0: Solution: \(\tan^{2}\left(x\right)-\frac{1}{\cos^{2}\left(x\right)}\) - step1: Reduce fractions to a common denominator: \(\frac{\tan^{2}\left(x\right)\cos^{2}\left(x\right)}{\cos^{2}\left(x\right)}-\frac{1}{\cos^{2}\left(x\right)}\) - step2: Transform the expression: \(\frac{\tan^{2}\left(x\right)\cos^{2}\left(x\right)-1}{\cos^{2}\left(x\right)}\) - step3: Transform the expression: \(\frac{\left(\tan\left(x\right)\cos\left(x\right)+1\right)\left(\tan\left(x\right)\cos\left(x\right)-1\right)}{\cos^{2}\left(x\right)}\) - step4: Transform the expression: \(\frac{\left(\sin\left(x\right)+1\right)\left(\sin\left(x\right)-1\right)}{\cos^{2}\left(x\right)}\) - step5: Transform the expression: \(\frac{-\cos^{2}\left(x\right)}{\cos^{2}\left(x\right)}\) - step6: Rewrite the fraction: \(-\frac{\cos^{2}\left(x\right)}{\cos^{2}\left(x\right)}\) - step7: Reduce the fraction: \(-1\) Calculate or simplify the expression \( (1/\tan(x) + \tan(x)) * (\sin(x) * \cos(x)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1}{\tan\left(x\right)}+\tan\left(x\right)\right)\left(\sin\left(x\right)\cos\left(x\right)\right)\) - step1: Remove the parentheses: \(\left(\frac{1}{\tan\left(x\right)}+\tan\left(x\right)\right)\sin\left(x\right)\cos\left(x\right)\) - step2: Add the terms: \(\frac{1+\tan^{2}\left(x\right)}{\tan\left(x\right)}\times \sin\left(x\right)\cos\left(x\right)\) - step3: Multiply the terms: \(\frac{\left(1+\tan^{2}\left(x\right)\right)\sin\left(x\right)\cos\left(x\right)}{\tan\left(x\right)}\) - step4: Transform the expression: \(\frac{\left(1+\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)^{2}\right)\sin\left(x\right)\cos\left(x\right)}{\tan\left(x\right)}\) - step5: Transform the expression: \(\frac{\left(1+\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)^{2}\right)\sin\left(x\right)\cos\left(x\right)}{\frac{\sin\left(x\right)}{\cos\left(x\right)}}\) - step6: Rewrite the expression: \(\frac{\frac{\left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\sin\left(x\right)\cos\left(x\right)}{\cos^{2}\left(x\right)}}{\frac{\sin\left(x\right)}{\cos\left(x\right)}}\) - step7: Multiply by the reciprocal: \(\frac{\left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\sin\left(x\right)\cos\left(x\right)}{\cos^{2}\left(x\right)}\times \frac{\cos\left(x\right)}{\sin\left(x\right)}\) - step8: Reduce the fraction: \(\frac{\left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\cos\left(x\right)}{\cos\left(x\right)}\times 1\) - step9: Multiply the terms: \(\frac{\left(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\right)\cos\left(x\right)}{\cos\left(x\right)}\) - step10: Reduce the fraction: \(\cos^{2}\left(x\right)+\sin^{2}\left(x\right)\) - step11: Transform the expression: \(1\) Calculate or simplify the expression \( \tan^2(x) * (1 - \sin^2(x)) \). Simplify the expression by following steps: - step0: Solution: \(\tan^{2}\left(x\right)\left(1-\sin^{2}\left(x\right)\right)\) - step1: Transform the expression: \(\tan^{2}\left(x\right)\cos^{2}\left(x\right)\) - step2: Multiply the terms: \(\left(\tan\left(x\right)\cos\left(x\right)\right)^{2}\) - step3: Transform the expression: \(\sin^{2}\left(x\right)\) Here are the simplified results for each expression: 1. **For** \( \tan^2(x) (1 - \sin^2(x)) \): \[ \tan^2(x) (1 - \sin^2(x)) = \sin^2(x) \] 2. **For** \( 8 \sin^2(\theta) + 8 \cos^2(\theta) \): \[ 8 \sin^2(\theta) + 8 \cos^2(\theta) = 8 \] 3. **For** \( (3 - 3 \sin(\theta))(3 + 3 \sin(\theta)) \): \[ (3 - 3 \sin(\theta))(3 + 3 \sin(\theta)) = 9 \cos^2(\theta) \] 4. **For** \( \left( \frac{1}{\tan(x)} + \tan(x) \right)(\sin(x) \cdot \cos(x)) \): \[ \left( \frac{1}{\tan(x)} + \tan(x) \right)(\sin(x) \cdot \cos(x)) = 1 \] 5. **For** \( \tan^2(x) - \frac{1}{\cos^2(x)} \): \[ \tan^2(x) - \frac{1}{\cos^2(x)} = -1 \] These results summarize the simplifications of the given expressions.