1. **Evaluate $\tan(\cos(-\frac{11\pi}{2}))$** - First, find $\cos(-\frac{11\pi}{2})$. - Since cosine is an even function, $\cos(-x) = \cos x$, so $\cos(-\frac{11\pi}{2}) = \cos(\frac{11\pi}{2})$. - Reduce $\frac{11\pi}{2}$ by subtracting multiples of $2\pi$: $$\frac{11\pi}{2} - 4\pi = \frac{11\pi}{2} - \frac{8\pi}{2} = \frac{3\pi}{2}$$ - So, $\cos(\frac{11\pi}{2}) = \cos(\frac{3\pi}{2}) = 0$. - Now, $\tan(0) = 0$. 2. **Find the exact value of $\sin(\frac{7\pi}{6})$** - $\frac{7\pi}{6}$ is in the third quadrant where sine is negative. - Reference angle is $\frac{7\pi}{6} - \pi = \frac{\pi}{6}$. - $\sin(\frac{\pi}{6}) = \frac{1}{2}$. - Therefore, $\sin(\frac{7\pi}{6}) = -\frac{1}{2}$. 3. **Simplify $\cot t \csc t (\sec^2 t - 1)$** - Recall the identity $\sec^2 t - 1 = \tan^2 t$. - Substitute: $\cot t \csc t \tan^2 t$. - Express in terms of sine and cosine: $$\cot t = \frac{\cos t}{\sin t}, \quad \csc t = \frac{1}{\sin t}, \quad \tan t = \frac{\sin t}{\cos t}$$ - So, $$\cot t \csc t \tan^2 t = \frac{\cos t}{\sin t} \cdot \frac{1}{\sin t} \cdot \left(\frac{\sin t}{\cos t}\right)^2 = \frac{\cos t}{\sin^2 t} \cdot \frac{\sin^2 t}{\cos^2 t} = \frac{\cos t}{\sin^2 t} \cdot \frac{\sin^2 t}{\cos^2 t}$$ - Simplify numerator and denominator: $$= \frac{\cos t}{\cos^2 t} = \frac{1}{\cos t} = \sec t$$ **Final answers:** - $\tan(\cos(-\frac{11\pi}{2})) = 0$ - $\sin(\frac{7\pi}{6}) = -\frac{1}{2}$ - $\cot t \csc t (\sec^2 t - 1) = \sec t$