Step 1: Simplify $\frac{\cos \theta}{\sin \theta}$. Using the quotient identity, we have: $$ \frac{\cos \theta}{\sin \theta} = \cot \theta $$ The correct option is 1. The final answer is $\boxed{\text{1) } \cot \theta}$. Step 2: Simplify $(1 + \tan^2 \theta)$. Using the Pythagorean identity, we have: $$ 1 + \tan^2 \theta = \sec^2 \theta $$ The correct option is 1. The final answer is $\boxed{\text{1) } \sec^2 \theta}$. Step 3: Find the value of $\cot(60^\circ)$. We know that $\tan(60^\circ) = \sqrt{3}$. Since $\cot \theta = \frac{1}{\tan \theta}$: $$ \cot(60^\circ) = \frac{1}{\tan(60^\circ)} = \frac{1}{\sqrt{3}} $$ The correct option is 3. The final answer is $\boxed{\text{3) } \frac{1}{\sqrt{3}}}$. Step 4: Find the value of $\sec(90^\circ)$. We know that $\sec \theta = \frac{1}{\cos \theta}$. Since $\cos(90^\circ) = 0$: $$ \sec(90^\circ) = \frac{1}{\cos(90^\circ)} = \frac{1}{0} $$ The value is undefined. The correct option is 3. The final answer is $\boxed{\text{3) undefined}}$. Step 5: If $\tan(\theta) = \sqrt{3}$, find $\theta$. We know that $\tan(60^\circ) = \sqrt{3}$. Therefore: $$ \theta = 60^\circ $$ The correct option is 3. The final answer is $\boxed{\text{3) } 60^\circ}$. Step 6: If $\sin \theta = \frac{3}{5}$ and $\theta$ is acute, find $\cos \theta$. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$: $$ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 $$ $$ \frac{9}{25} + \cos^2 \theta = 1 $$ $$ \cos^2 \theta = 1 - \frac{9}{25} $$ $$ \cos^2 \theta = \frac{25 - 9}{25} $$ $$ \cos^2 \theta = \frac{16}{25} $$ $$ \cos \theta = \pm \sqrt{\frac{16}{25}} $$ $$ \cos \theta = \pm \frac{4}{5} $$ Since $\theta$ is acute (in the first quadrant), $\cos \theta$ must be positive. $$ \cos \theta = \frac{4}{5} $$ The correct option is 1. The final answer is $\boxed{\text{1) } \frac{4}{5}}$. Step 7: If $\sin \theta - \cos \theta = 0$, find $\theta$. Given the equation: $$ \sin \theta - \cos \theta = 0 $$ $$ \sin \theta = \cos \theta $$ Divide both sides by $\cos \theta$ (assuming $\cos \theta \neq 0$): $$ \frac{\sin \theta}{\cos \theta} = 1 $$ $$ \tan \theta = 1 $$ We know that $\tan(45^\circ) = 1$. Therefore: $$ \theta = 45^\circ $$ The correct option is 2. The final answer is $\boxed{\text{2) } 45^\circ}$. Step 8: If $(\csc \theta - \cot \theta) = x$, find $(\csc \theta + \cot \theta)$. Using the Pythagorean identity $\csc^2 \theta - \cot^2 \theta = 1$. Factor the left side as a difference of squares: $$ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = 1 $$ Substitute the given value $(\csc \theta - \cot \theta) = x$: $$ x(\csc \theta + \cot \theta) = 1 $$ Solve for $(\csc \theta + \cot \theta)$: $$ \csc \theta + \cot \theta = \frac{1}{x} $$ The correct option is 1. The final answer is $\boxed{\text{1) } \frac{1}{x}}$. Step 9: Find the value of $\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}$. This is the double angle formula for cosine: $$ \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos(2\theta) $$ The correct option is 1. The final answer is $\boxed{\text{1) } \cos(2\theta)}$. Send me the next one 📸