If sin `theta = 4/5`, then value of cos `2 theta` is

Options

A. `8//5`
B. `3//5`
C. `7//35`
D. `-7//25`

Answer

To find the value of cos(2θ) given that sin(θ) = 4/5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given value:**
We know that sin(θ) = 4/5.

2. **Calculate sin²(θ):**
\[ \sin^2(θ) = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

3. **Use the Pythagorean identity to find cos²(θ):**
We know that:
\[ \sin^2(θ) + \cos^2(θ) = 1 \]
Substituting the value of sin²(θ):
\[ \frac{16}{25} + \cos^2(θ) = 1 \]
Rearranging gives:
\[ \cos^2(θ) = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

4. **Calculate cos(2θ) using the double angle formula:**
The formula for cos(2θ) is:
\[ \cos(2θ) = \cos^2(θ) - \sin^2(θ) \]
Substituting the values we found:
\[ \cos(2θ) = \frac{9}{25} - \frac{16}{25} \]
Simplifying this gives:
\[ \cos(2θ) = \frac{9 - 16}{25} = \frac{-7}{25} \]

5. **Final answer:**
Therefore, the value of cos(2θ) is:
\[ \cos(2θ) = -\frac{7}{25} \]