`( 2 sec theta + 3 tan theta + 5 sin theta - 7 cos theta +5)/( 2 tan theta + 3 sec theta +5 cos theta + 7 sin theta + 8 )=`

Options

A. `tan"" ( theta)/( 2)`
B. `cot "" ( theta )/( 2)`
C. `sec "" ( theta )/( 2)`
D. `cosec "" ( theta )/( 2)`

Answer

To solve the expression

\[ \frac{2 \sec \theta + 3 \tan \theta + 5 \sin \theta - 7 \cos \theta + 5}{2 \tan \theta + 3 \sec \theta + 5 \cos \theta + 7 \sin \theta + 8} \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Rewrite Secant and Tangent in Terms of Sine and Cosine
Recall that:
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Substituting these into the expression gives:

\[ \frac{2 \cdot \frac{1}{\cos \theta} + 3 \cdot \frac{\sin \theta}{\cos \theta} + 5 \sin \theta - 7 \cos \theta + 5}{2 \cdot \frac{\sin \theta}{\cos \theta} + 3 \cdot \frac{1}{\cos \theta} + 5 \cos \theta + 7 \sin \theta + 8} \]

### Step 2: Combine Terms in the Numerator
The numerator becomes:

\[ \frac{2 + 3 \sin \theta + 5 \sin \theta \cos \theta - 7 \cos^2 \theta + 5 \cos \theta}{\cos \theta} \]

Combining like terms gives:

\[ \frac{(2 + 5) + (3 + 5) \sin \theta - 7 \cos^2 \theta}{\cos \theta} = \frac{7 + 8 \sin \theta - 7 \cos^2 \theta}{\cos \theta} \]

### Step 3: Combine Terms in the Denominator
The denominator becomes:

\[ \frac{2 \sin \theta + 3 + 5 \cos^2 \theta + 7 \sin \theta + 8}{\cos \theta} \]

Combining like terms gives:

\[ \frac{(3 + 8) + (2 + 7) \sin \theta + 5 \cos^2 \theta}{\cos \theta} = \frac{11 + 9 \sin \theta + 5 \cos^2 \theta}{\cos \theta} \]

### Step 4: Simplify the Entire Expression
Now, substituting the simplified numerator and denominator back into the expression gives:

\[ \frac{7 + 8 \sin \theta - 7 \cos^2 \theta}{11 + 9 \sin \theta + 5 \cos^2 \theta} \]

### Step 5: Factor and Simplify Further
Notice that \( \cos^2 \theta = 1 - \sin^2 \theta \). Substituting this gives:

Numerator:
\[ 7 + 8 \sin \theta - 7(1 - \sin^2 \theta) = 7 + 8 \sin \theta - 7 + 7 \sin^2 \theta = 8 \sin \theta + 7 \sin^2 \theta \]

Denominator:
\[ 11 + 9 \sin \theta + 5(1 - \sin^2 \theta) = 11 + 9 \sin \theta + 5 - 5 \sin^2 \theta = 16 + 9 \sin \theta - 5 \sin^2 \theta \]

### Final Expression
Thus, the expression simplifies to:

\[ \frac{7 \sin^2 \theta + 8 \sin \theta}{-5 \sin^2 \theta + 9 \sin \theta + 16} \]

### Conclusion
This expression can be further analyzed or evaluated depending on the value of \(\theta\).

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