Without using trigonometric tables , evaluate :
(i) `(sin11^(@))/(cos79^(@))`
(ii)`(sec15^(@))/("cosec"75^(@))`
(iii) `(tan54^(@))/(cot36^(@))`
(iv)` (cos68^(@))/(sin22^(@))`
(v) `("cosec"24^(@))/(sec66^(@))`
(vi) `(cot18^(@))/(tan72^(@))`

Answer

To evaluate the given expressions without using trigonometric tables, we can use trigonometric identities and co-function relationships. Let's solve each part step by step.

### (i) Evaluate \( \frac{\sin 11^\circ}{\cos 79^\circ} \)

1. **Use the co-function identity**: We know that \( \sin \theta = \cos(90^\circ - \theta) \).
- Here, \( \sin 11^\circ = \cos(90^\circ - 11^\circ) = \cos 79^\circ \).

2. **Substituting the identity**:
\[ \frac{\sin 11^\circ}{\cos 79^\circ} = \frac{\cos 79^\circ}{\cos 79^\circ} = 1 \]

### (ii) Evaluate \( \frac{\sec 15^\circ}{\csc 75^\circ} \)

1. **Use the identities**: We know that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).
- Thus, \( \frac{\sec 15^\circ}{\csc 75^\circ} = \frac{1/\cos 15^\circ}{1/\sin 75^\circ} = \frac{\sin 75^\circ}{\cos 15^\circ} \).

2. **Use the co-function identity**: \( \sin 75^\circ = \cos(90^\circ - 75^\circ) = \cos 15^\circ \).

3. **Substituting the identity**:
\[ \frac{\sin 75^\circ}{\cos 15^\circ} = \frac{\cos 15^\circ}{\cos 15^\circ} = 1 \]

### (iii) Evaluate \( \frac{\tan 54^\circ}{\cot 36^\circ} \)

1. **Use the identity**: We know that \( \cot \theta = \frac{1}{\tan \theta} \).
- Thus, \( \frac{\tan 54^\circ}{\cot 36^\circ} = \tan 54^\circ \cdot \tan 36^\circ \).

2. **Use the co-function identity**: \( \tan 54^\circ = \cot(90^\circ - 54^\circ) = \cot 36^\circ \).

3. **Substituting the identity**:
\[ \frac{\tan 54^\circ}{\cot 36^\circ} = \frac{\cot 36^\circ}{\cot 36^\circ} = 1 \]

### (iv) Evaluate \( \frac{\cos 68^\circ}{\sin 22^\circ} \)

1. **Use the co-function identity**: \( \sin 22^\circ = \cos(90^\circ - 22^\circ) = \cos 68^\circ \).

2. **Substituting the identity**:
\[ \frac{\cos 68^\circ}{\sin 22^\circ} = \frac{\cos 68^\circ}{\cos 68^\circ} = 1 \]

### (v) Evaluate \( \frac{\csc 24^\circ}{\sec 66^\circ} \)

1. **Use the identities**: We know that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).
- Thus, \( \frac{\csc 24^\circ}{\sec 66^\circ} = \frac{1/\sin 24^\circ}{1/\cos 66^\circ} = \frac{\cos 66^\circ}{\sin 24^\circ} \).

2. **Use the co-function identity**: \( \cos 66^\circ = \sin(90^\circ - 66^\circ) = \sin 24^\circ \).

3. **Substituting the identity**:
\[ \frac{\cos 66^\circ}{\sin 24^\circ} = \frac{\sin 24^\circ}{\sin 24^\circ} = 1 \]

### (vi) Evaluate \( \frac{\cot 18^\circ}{\tan 72^\circ} \)

1. **Use the identity**: We know that \( \tan \theta = \frac{1}{\cot \theta} \).
- Thus, \( \frac{\cot 18^\circ}{\tan 72^\circ} = \cot 18^\circ \cdot \cot 72^\circ \).

2. **Use the co-function identity**: \( \tan 72^\circ = \cot(90^\circ - 72^\circ) = \cot 18^\circ \).

3. **Substituting the identity**:
\[ \frac{\cot 18^\circ}{\tan 72^\circ} = \frac{\cot 18^\circ}{\cot 18^\circ} = 1 \]

### Summary of Results:
- (i) \( 1 \)
- (ii) \( 1 \)
- (iii) \( 1 \)
- (iv) \( 1 \)
- (v) \( 1 \)
- (vi) \( 1 \)