The value of
`(cos (90^(@) + A) sec ( 360^(@) - A) tan ( 180^(@) - A))/( sec (A - 720^(@)) sin ( 54 0 ^(@) + A) cot (A - 90^(@)))` is

Options

A. 0
B. 1
C. `oo`
D. None of these

Answer

To solve the expression

\[ \frac{\cos(90^\circ + A) \sec(360^\circ - A) \tan(180^\circ - A)}{\sec(A - 720^\circ) \sin(540^\circ + A) \cot(A - 90^\circ)} \]

we will simplify each trigonometric function step by step.

### Step 1: Simplifying \(\cos(90^\circ + A)\)

Using the identity \(\cos(90^\circ + \theta) = -\sin(\theta)\):

\[ \cos(90^\circ + A) = -\sin(A) \]

### Step 2: Simplifying \(\sec(360^\circ - A)\)

Using the identity \(\sec(360^\circ - \theta) = \sec(\theta)\):

\[ \sec(360^\circ - A) = \sec(A) \]

### Step 3: Simplifying \(\tan(180^\circ - A)\)

Using the identity \(\tan(180^\circ - \theta) = -\tan(\theta)\):

\[ \tan(180^\circ - A) = -\tan(A) \]

### Step 4: Putting it all together in the numerator

Now substituting these results into the numerator:

\[ \text{Numerator} = (-\sin(A)) \cdot \sec(A) \cdot (-\tan(A)) = \sin(A) \sec(A) \tan(A) \]

### Step 5: Simplifying \(\sec(A - 720^\circ)\)

Using the identity \(\sec(\theta - 360^\circ) = \sec(\theta)\):

\[ \sec(A - 720^\circ) = \sec(A) \]

### Step 6: Simplifying \(\sin(540^\circ + A)\)

Using the identity \(\sin(540^\circ + \theta) = -\sin(\theta)\):

\[ \sin(540^\circ + A) = -\sin(A) \]

### Step 7: Simplifying \(\cot(A - 90^\circ)\)

Using the identity \(\cot(\theta - 90^\circ) = -\tan(\theta)\):

\[ \cot(A - 90^\circ) = -\tan(A) \]

### Step 8: Putting it all together in the denominator

Now substituting these results into the denominator:

\[ \text{Denominator} = \sec(A) \cdot (-\sin(A)) \cdot (-\tan(A)) = \sec(A) \sin(A) \tan(A) \]

### Step 9: Final simplification

Now we have:

\[ \frac{\sin(A) \sec(A) \tan(A)}{\sec(A) \sin(A) \tan(A)} \]

Both the numerator and denominator are equal, thus simplifying to:

\[ \frac{\sin(A) \sec(A) \tan(A)}{\sin(A) \sec(A) \tan(A)} = 1 \]

### Final Answer

The value of the expression is:

\[ \boxed{1} \]