The value of `(cos^(n)38^(@)-cot^(n)52^(@))/(sin^(n)52^(@)-tan^(n)38^(@))=?`

Options

A. 5
B. 7
C. 1
D. 3

Answer

To solve the expression \((\cos^n 38^\circ - \cot^n 52^\circ) / (\sin^n 52^\circ - \tan^n 38^\circ)\), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We can use the identities:
- \(\cos \theta = \sin(90^\circ - \theta)\)
- \(\cot \theta = \tan(90^\circ - \theta)\)

Applying these identities, we can rewrite the terms in the expression.

### Step 2: Apply the identities
1. Rewrite \(\cos 38^\circ\):
\[ \cos 38^\circ = \sin(90^\circ - 38^\circ) = \sin 52^\circ \]
2. Rewrite \(\cot 52^\circ\):
\[ \cot 52^\circ = \tan(90^\circ - 52^\circ) = \tan 38^\circ \]

### Step 3: Substitute back into the expression
Now substitute these identities into the original expression:
\[ \frac{\cos^n 38^\circ - \cot^n 52^\circ}{\sin^n 52^\circ - \tan^n 38^\circ} = \frac{\sin^n 52^\circ - \tan^n 38^\circ}{\sin^n 52^\circ - \tan^n 38^\circ} \]

### Step 4: Simplify the expression
Since the numerator and denominator are identical, they cancel each other out:
\[ \frac{\sin^n 52^\circ - \tan^n 38^\circ}{\sin^n 52^\circ - \tan^n 38^\circ} = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]

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