Evaluating Trigonometric Expressions

The question asks us to evaluate the trigonometric expression: $\sin 25^\circ \sin 65^\circ - \cos 25^\circ \cos 65^\circ$.

This expression involves products of sine and cosine functions of two different angles, $25^\circ$ and $65^\circ$. We can approach this problem by recognizing a standard trigonometric identity or by using complementary angle relationships.

Using Trigonometric Identities (Angle Sum Identity)

Let's rearrange the given expression:

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

We can factor out -1:

Expression $= -(\cos 25^\circ \cos 65^\circ - \sin 25^\circ \sin 65^\circ)$

Now, let's recall the angle sum identity for cosine:

• $\cos(A+B) = \cos A \cos B - \sin A \sin B$

Comparing this identity with the expression inside the parenthesis, we can see that it matches the right side of the identity with $A = 25^\circ$ and $B = 65^\circ$.

So, $\cos 25^\circ \cos 65^\circ - \sin 25^\circ \sin 65^\circ = \cos(25^\circ + 65^\circ)$.

Let's calculate the sum of the angles:

$25^\circ + 65^\circ = 90^\circ$

Therefore, $\cos(25^\circ + 65^\circ) = \cos 90^\circ$.

We know the standard value of $\cos 90^\circ$:

• $\cos 90^\circ = 0$

Substituting this back into our expression:

Expression $= -(\cos(25^\circ + 65^\circ))$

Expression $= -(\cos 90^\circ)$

Expression $= -(0)$

Expression $= 0$

Using Complementary Angle Relationships

Alternatively, we can use the complementary angle relationships. Notice that $25^\circ + 65^\circ = 90^\circ$. This means $65^\circ$ is the complement of $25^\circ$, i.e., $65^\circ = 90^\circ - 25^\circ$. Similarly, $25^\circ = 90^\circ - 65^\circ$.

The complementary angle identities are:

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

Let's apply these to the $65^\circ$ terms in the expression:

• $\sin 65^\circ = \sin(90^\circ - 25^\circ) = \cos 25^\circ$
• $\cos 65^\circ = \cos(90^\circ - 25^\circ) = \sin 25^\circ$

Now substitute these into the original expression:

Expression $= \sin 25^\circ (\sin 65^\circ) - \cos 25^\circ (\cos 65^\circ)$

Substitute the converted terms:

Expression $= \sin 25^\circ (\cos 25^\circ) - \cos 25^\circ (\sin 25^\circ)$

This simplifies to:

Expression $= \sin 25^\circ \cos 25^\circ - \sin 25^\circ \cos 25^\circ$

This is the difference of a term and itself, which is always 0.

Expression $= 0$

Conclusion on Evaluating the Expression

Both methods demonstrate that the value of the expression $\sin 25^\circ \sin 65^\circ - \cos 25^\circ \cos 65^\circ$ is 0.

Revision Table: Key Trigonometric Concepts

Additional Information on Trigonometry

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The trigonometric functions (sine, cosine, tangent, etc.) are periodic functions used to model various cyclic phenomena.

Understanding trigonometric identities is crucial for simplifying expressions and solving trigonometric equations. Identities like the angle sum and difference formulas, double angle formulas, half angle formulas, and product-to-sum formulas are widely used.

Complementary angles are two angles that add up to $90^\circ$. The relationships between trigonometric functions of complementary angles are fundamental and often help simplify expressions involving angles like $10^\circ$ and $80^\circ$, $30^\circ$ and $60^\circ$, etc.

Knowing the standard values of trigonometric functions for specific angles (like $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$) is also essential for evaluating expressions.