`7cosectheta+24sectheta=25cosecthetasectheta`, then `costheta=?`

Options

A. `(5)/(24)`
B. `(7)/(24)`
C. `(11)/(24)`
D. `(13)/(24)`

Answer

To solve the equation \( 7 \csc \theta + 24 \sec \theta = 25 \csc \theta \sec \theta \) and find the value of \( \cos \theta \), we can follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that:
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \sec \theta = \frac{1}{\cos \theta} \)

Substituting these into the equation gives us:
\[ 7 \left(\frac{1}{\sin \theta}\right) + 24 \left(\frac{1}{\cos \theta}\right) = 25 \left(\frac{1}{\sin \theta}\right) \left(\frac{1}{\cos \theta}\right) \]

### Step 2: Clear the fractions
To eliminate the fractions, multiply through by \( \sin \theta \cos \theta \):
\[ 7 \cos \theta + 24 \sin \theta = 25 \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ 7 \cos \theta + 24 \sin \theta - 25 = 0 \]

### Step 4: Isolate one of the trigonometric functions
We can express \( \sin \theta \) in terms of \( \cos \theta \):
\[ 24 \sin \theta = 25 - 7 \cos \theta \]
\[ \sin \theta = \frac{25 - 7 \cos \theta}{24} \]

### Step 5: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \left(\frac{25 - 7 \cos \theta}{24}\right)^2 + \cos^2 \theta = 1 \]

### Step 6: Expand and simplify
Expanding the left side:
\[ \frac{(25 - 7 \cos \theta)^2}{576} + \cos^2 \theta = 1 \]
Multiply through by 576 to eliminate the fraction:
\[ (25 - 7 \cos \theta)^2 + 576 \cos^2 \theta = 576 \]

### Step 7: Expand the squared term
Expanding \( (25 - 7 \cos \theta)^2 \):
\[ 625 - 350 \cos \theta + 49 \cos^2 \theta + 576 \cos^2 \theta = 576 \]
Combine like terms:
\[ 625 - 350 \cos \theta + 625 \cos^2 \theta = 576 \]

### Step 8: Rearrange to form a quadratic equation
Rearranging gives:
\[ 625 \cos^2 \theta - 350 \cos \theta + 49 = 0 \]

### Step 9: Use the quadratic formula
Using the quadratic formula \( \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 625, b = -350, c = 49 \):
\[ \cos \theta = \frac{350 \pm \sqrt{(-350)^2 - 4 \cdot 625 \cdot 49}}{2 \cdot 625} \]
Calculate the discriminant:
\[ \sqrt{122500 - 122500} = 0 \]
Thus:
\[ \cos \theta = \frac{350}{1250} = \frac{7}{25} \]

### Final Answer
\[ \cos \theta = \frac{7}{25} \]

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