Evaluating the Trigonometric Expression

We need to find the value of the given trigonometric expression:

$$ \frac{\cos 25^\circ - \sin 65^\circ}{\cos 25^\circ + \sin 65^\circ} $$

Using Trigonometric Identities

To simplify this expression, we can use trigonometric identities. One useful identity relates sine and cosine of complementary angles (angles that add up to 90 degrees). The identity is:

$$ \sin(90^\circ - \theta) = \cos \theta $$

We can apply this identity to the term $\sin 65^\circ$ in the expression. Let's rewrite $65^\circ$ as $(90^\circ - 25^\circ)$.

So, $$ \sin 65^\circ = \sin(90^\circ - 25^\circ) $$

Using the identity $\sin(90^\circ - \theta) = \cos \theta$, we get:

$$ \sin 65^\circ = \cos 25^\circ $$

Substituting and Simplifying

Now, substitute this result back into the original expression:

$$ \frac{\cos 25^\circ - \sin 65^\circ}{\cos 25^\circ + \sin 65^\circ} = \frac{\cos 25^\circ - \cos 25^\circ}{\cos 25^\circ + \cos 25^\circ} $$

Let's simplify the numerator and the denominator separately.

• Numerator: $ \cos 25^\circ - \cos 25^\circ = 0 $
• Denominator: $ \cos 25^\circ + \cos 25^\circ = 2 \cos 25^\circ $

So the expression becomes:

$$ \frac{0}{2 \cos 25^\circ} $$

Calculating the Final Value

Since the numerator is 0 and the denominator ($2 \cos 25^\circ$) is not zero (as $\cos 25^\circ \neq 0$), the value of the fraction is 0.

$$ \frac{0}{2 \cos 25^\circ} = 0 $$

Conclusion

The value of the expression $\frac{\cos 25^\circ - \sin 65^\circ}{\cos 25^\circ + \sin 65^\circ}$ is 0.