Simplif. \( \tan ^{2} x\left(1-\sin ^{2} x\right) \) 2. \( (3-3 \sin \theta)(3+3 \sin \theta) \) 3. \( \left(\frac{1}{\tan x}+\tan x\right) \cdot(\sin x \cdot \cos x) \) Prii! 4. \( \cos ^{3} x+\cos x \cdot \sin ^{2} x=\cos x \) 5. \( \tan ^{2} \theta+1=\frac{1}{\cos ^{2} \theta} \)
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First additional information: The expression \( \tan^{2} x(1-\sin^{2} x) \) can be simplified using the Pythagorean identity \( \tan^{2} x = \frac{\sin^{2} x}{\cos^{2} x} \) and recognizing that \( 1 - \sin^{2} x = \cos^{2} x \). This leads to the simplification \( \tan^{2} x \cdot \cos^{2} x = \sin^{2} x \), highlighting the interconnectedness of trigonometric identities! Second additional information: The expression \( (3 - 3\sin \theta)(3 + 3\sin \theta) \) is a classic difference of squares that simplifies to \( 9 - 9\sin^{2} \theta = 9(1 - \sin^2 \theta) \). By employing the identity \( 1 - \sin^{2} \theta = \cos^{2} \theta \), it becomes \( 9 \cos^{2} \theta \), demonstrating how basic algebraic techniques can reveal deeper trigonometric relationships!