Solve \( \frac{\cos 33^{\circ}}{\sin 54^{\circ}}+\frac{\sec 79^{\circ}}{\operatorname{cosec} 11^{\circ}}+\cos ^{2} 45^{\circ} \)

To solve the expression \[ \frac{\cos 33^{\circ}}{\sin 54^{\circ}}+\frac{\sec 79^{\circ}}{\operatorname{cosec} 11^{\circ}}+\cos ^{2} 45^{\circ}, \] we'll simplify each term step by step. 1. **Simplifying the first term:** We know that \( \sin 54^{\circ} = \cos(90^{\circ} - 54^{\circ}) = \cos 36^{\circ} \). Therefore, we can rewrite the first term: \[ \frac{\cos 33^{\circ}}{\sin 54^{\circ}} = \frac{\cos 33^{\circ}}{\cos 36^{\circ}}. \] 2. **Simplifying the second term:** The second term involves secant and cosecant. We rewrite it using notations of sine and cosine: \[ \frac{\sec 79^{\circ}}{\operatorname{cosec} 11^{\circ}} = \frac{1/\cos 79^{\circ}}{1/\sin 11^{\circ}} = \frac{\sin 11^{\circ}}{\cos 79^{\circ}}. \] Since \( \cos 79^{\circ} = \sin(90^{\circ} - 79^{\circ}) = \sin 11^{\circ} \), we have: \[ \frac{\sin 11^{\circ}}{\cos 79^{\circ}} = \frac{\sin 11^{\circ}}{\sin 11^{\circ}} = 1. \] 3. **Simplifying the third term:** For the last term, we have: \[ \cos^2 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}. \] 4. **Combining the results:** Adding all the simplified parts together, we have: \[ \frac{\cos 33^{\circ}}{\cos 36^{\circ}} + 1 + \frac{1}{2}. \] To evaluate \( \frac{\cos 33^{\circ}}{\cos 36^{\circ}} \), note that this expression isn't easily simplified without a calculator—but we can recognize that \( \cos 33^{\circ} \approx 0.8446 \) and \( \cos 36^{\circ} \approx 0.8090 \). Hence: \[ \frac{\cos 33^{\circ}}{\cos 36^{\circ}} \approx \frac{0.8446}{0.8090} \approx 1.0439. \] Thus, combining all these values: \[ 1.0439 + 1 + 0.5 \approx 2.5439. \] Finally, we can round our final answer: \[ \frac{\cos 33^{\circ}}{\cos 36^{\circ}} + 1 + \frac{1}{2} \approx 2.54. \] The final answer is: \[ \frac{\cos 33^{\circ}}{\sin 54^{\circ}}+\frac{\sec 79^{\circ}}{\operatorname{cosec} 11^{\circ}}+\cos ^{2} 45^{\circ} \approx 2.54. \]