1. **Find \( \tan\left(-\frac{2\pi}{3}\right) \)** Step 1. Recognize the angle \( -\frac{2\pi}{3} \) is negative. Add \( 2\pi \) to find a positive coterminal angle: $$ -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3} $$ Step 2. \( \tan\left(-\frac{2\pi}{3}\right) = \tan\left(\frac{4\pi}{3}\right) \). Step 3. \( \frac{4\pi}{3} \) is in the third quadrant where tangent is positive. Step 4. Reference angle is \( \pi - \frac{4\pi}{3} = \pi - \frac{4\pi}{3} = -\frac{\pi}{3} \), but taking absolute value gives \( \frac{\pi}{3} \). Step 5. \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \), so \[ \tan\left(-\frac{2\pi}{3}\right) = \sqrt{3} \] 2. **Find \( \sec\left(\frac{3\pi}{4}\right) \)** Step 1. \( \sec \theta = \frac{1}{\cos \theta} \). Step 2. \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) since \( \frac{3\pi}{4} \) is in quadrant II where cosine is negative. Step 3. Thus, \[ \sec\left(\frac{3\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \] 3. **Find \( \csc 630^\circ \)** Step 1. Subtract \( 360^\circ \) to find coterminal angle within \( 0^\circ - 360^\circ \): \[ 630^\circ - 360^\circ = 270^\circ \] Step 2. \( \csc 630^\circ = \csc 270^\circ = \frac{1}{\sin 270^\circ} \) Step 3. \( \sin 270^\circ = -1 \) Step 4. So, \[ \csc 630^\circ = \frac{1}{-1} = -1 \] 4. **Find \( \cot(-720^\circ) \)** Step 1. Add \( 720^\circ \) (two full rotations) since \( \cot(\theta) \) is periodic with period \( 180^\circ \), but we can also add \( 360^\circ \) multiples. \[ -720^\circ + 720^\circ = 0^\circ \] Step 2. \( \cot(-720^\circ) = \cot 0^\circ = \frac{\cos 0^\circ}{\sin 0^\circ} = \frac{1}{0} \) which is undefined. 5. **Find \( \sin 45^\circ + 3\cos 90^\circ \)** Step 1. \( \sin 45^\circ = \frac{\sqrt{2}}{2} \) Step 2. \( \cos 90^\circ = 0 \) Step 3. Compute expression: \[ \sin 45^\circ + 3 \cos 90^\circ = \frac{\sqrt{2}}{2} + 3 \cdot 0 = \frac{\sqrt{2}}{2} \] 6. **Find \( (\sec 135^\circ)(\cot 135^\circ) \)** Step 1. \( \cos 135^\circ = -\frac{\sqrt{2}}{2} \) (quadrant II) Step 2. \( \sec 135^\circ = \frac{1}{\cos 135^\circ} = -\sqrt{2} \) Step 3. \( \sin 135^\circ = \frac{\sqrt{2}}{2} \) Step 4. \( \cot 135^\circ = \frac{\cos 135^\circ}{\sin 135^\circ} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 \) Step 5. Multiply: \[ (\sec 135^\circ)(\cot 135^\circ) = (-\sqrt{2})(-1) = \sqrt{2} \] **Final Answers:** 1. \( \tan\left(-\frac{2\pi}{3}\right) = \sqrt{3} \) 2. \( \sec\left(\frac{3\pi}{4}\right) = -\sqrt{2} \) 3. \( \csc 630^\circ = -1 \) 4. \( \cot(-720^\circ) \) is undefined 5. \( \sin 45^\circ + 3 \cos 90^\circ = \frac{\sqrt{2}}{2} \) 6. \( (\sec 135^\circ)(\cot 135^\circ) = \sqrt{2} \)