Prove the following : sin25*cos5-cos80*sin70=sin15*cos5

To prove the equation \( \sin 25^\circ \cos 5^\circ - \cos 80^\circ \sin 70^\circ = \sin 15^\circ \cos 5^\circ \), we can use some trigonometric identities and properties. The main identity that will be useful here is the sine subtraction formula, which states that \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). This can help us relate different sine and cosine values.

Breaking Down the Left Side

Let’s start with the left side of the equation:

Step 1: Rewrite \( \cos 80^\circ \) and \( \sin 70^\circ \) using complementary angles.

• Recall that \( \cos(90^\circ - x) = \sin x \). So, \( \cos 80^\circ = \sin 10^\circ \) and \( \sin 70^\circ = \cos 20^\circ \).

Substituting these values gives us:

\( \sin 25^\circ \cos 5^\circ - \sin 10^\circ \cos 20^\circ \)

Step 2: Now, apply the sine subtraction formula:

We can rewrite \( \sin 25^\circ \cos 5^\circ - \sin 10^\circ \cos 20^\circ \) as follows:

\( \sin(25^\circ - 10^\circ) \cos 5^\circ = \sin 15^\circ \cos 5^\circ \)

This shows that the left side simplifies directly to \( \sin 15^\circ \cos 5^\circ \).

Examining the Right Side

Now let’s look at the right side of the original equation:

We have \( \sin 15^\circ \cos 5^\circ \), which matches exactly what we derived from the left side after simplification.

Final Verification

Since both sides are equal, we have successfully proven that:

Result: \( \sin 25^\circ \cos 5^\circ - \cos 80^\circ \sin 70^\circ = \sin 15^\circ \cos 5^\circ \)

This demonstrates the power of using trigonometric identities to simplify expressions effectively. If you have any further questions about trigonometric identities or how to apply them, feel free to ask!