Question

\(\cos(36^\circ) \cos(54^\circ) - \sin(36^\circ) \sin(54^\circ)=\)

• A. 1
• B. 0
• C. –1
• D. \(\frac{1}{2}\)

Answer 1

The Correct Option is B

Given Expression: 

\[ \cos(36^\circ) \cos(54^\circ) - \sin(36^\circ) \sin(54^\circ) \]

Step 1: Use the Cosine Addition Identity

We know that: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] Comparing with our expression, we see that: \[ \cos(36^\circ + 54^\circ) = \cos(90^\circ) \]

Step 2: Compute \( \cos(90^\circ) \)

\[ \cos(90^\circ) = 0 \]

Final Answer: 0

Answer 2

To solve the given expression \(\cos(36^\circ) \cos(54^\circ) - \sin(36^\circ) \sin(54^\circ)\), we can use the trigonometric identity for cosine of a sum: \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] By comparing the given expression with the identity, we note that it matches the form of \(\cos(A+B)\) where \(A=36^\circ\) and \(B=54^\circ\). 
Therefore, the given expression simplifies to: \[ \cos(36^\circ + 54^\circ) \] Calculating \(36^\circ + 54^\circ\) gives us \(90^\circ\). 
Therefore, the expression becomes: \[ \cos(90^\circ) \] Since \(\cos(90^\circ)=0\), the value of the original expression is \(0\). 
Thus, the correct answer is: 0