Question

The value of sin34°+cos64°-cos4° is______

• A. 0
• B. 3
• C. 2
• D. 1

Answer 1

The Correct Option is A

We need to evaluate the expression \( \sin 34^\circ + \cos 64^\circ - \cos 4^\circ \).

1. Simplify Using Trigonometric Identities:
Notice the angles: \( 64^\circ = 90^\circ - 26^\circ \), and we can explore relationships with \( 34^\circ \) and \( 4^\circ \). Use the co-function identity for cosine:

\( \cos 64^\circ = \cos (90^\circ - 26^\circ) = \sin 26^\circ \)
The expression becomes:

\( \sin 34^\circ + \sin 26^\circ - \cos 4^\circ \)

2. Examine the Angles:
Notice that \( 34^\circ - 26^\circ = 8^\circ \), and \( 4^\circ \times 2 = 8^\circ \), suggesting a possible relationship. Let’s try to use sum-to-product identities for the sine terms:

\( \sin 34^\circ + \sin 26^\circ \)
Use the identity \( \sin a + \sin b = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right) \), where \( a = 34^\circ \), \( b = 26^\circ \):

\( \frac{a+b}{2} = \frac{34^\circ + 26^\circ}{2} = 30^\circ \), \( \frac{a-b}{2} = \frac{34^\circ - 26^\circ}{2} = 4^\circ \)
Thus:

\( \sin 34^\circ + \sin 26^\circ = 2 \sin 30^\circ \cos 4^\circ \)
Since \( \sin 30^\circ = \frac{1}{2} \), we get:

\( 2 \sin 30^\circ \cos 4^\circ = 2 \cdot \frac{1}{2} \cdot \cos 4^\circ = \cos 4^\circ \)

3. Substitute Back:
The original expression is now:

\( \sin 34^\circ + \cos 64^\circ - \cos 4^\circ = (\sin 34^\circ + \sin 26^\circ) - \cos 4^\circ \)
\( = \cos 4^\circ - \cos 4^\circ = 0 \)

4. Verify the Result:
The cancellation suggests the expression simplifies to zero. To confirm, we could compute numerically, but the algebraic simplification is consistent, and the angles’ relationships support this outcome.

Final Answer:
The value of \( \sin 34^\circ + \cos 64^\circ - \cos 4^\circ \) is \( 0 \).