Question:
The value of $\sin6^\circ \cos36^\circ \sin66^\circ + \cos12^\circ \sin42^\circ \sin18^\circ$ is equal to:
The value of $\sin6^\circ \cos36^\circ \sin66^\circ + \cos12^\circ \sin42^\circ \sin18^\circ$ is equal to:
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Use product-to-sum identities to simplify multiple trig product terms.
Updated On: Apr 24, 2026
- $\frac{1}{12}(\sin18^\circ + \cos36^\circ)$
- $\frac{1}{3}(\sin18^\circ + \cos36^\circ)$
- $\frac{1}{16}(\sin18^\circ + \cos36^\circ)$
- $\frac{1}{4}(\sin18^\circ + \cos36^\circ)$
- $\frac{1}{2}(\sin18^\circ + \cos36^\circ)$
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The Correct Option is D
Solution and Explanation
Concept:
• Use product-to-sum identities: \[ \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \]
Step 1: Group terms cleverly
Use identities to simplify each product. After applying identities and simplifying: \[ \sin6^\circ \cos36^\circ \sin66^\circ + \cos12^\circ \sin42^\circ \sin18^\circ = \frac{1}{4}(\sin18^\circ + \cos36^\circ) \] Final Conclusion:
\[ = \frac{1}{4}(\sin18^\circ + \cos36^\circ) \]
• Use product-to-sum identities: \[ \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \]
Step 1: Group terms cleverly
Use identities to simplify each product. After applying identities and simplifying: \[ \sin6^\circ \cos36^\circ \sin66^\circ + \cos12^\circ \sin42^\circ \sin18^\circ = \frac{1}{4}(\sin18^\circ + \cos36^\circ) \] Final Conclusion:
\[ = \frac{1}{4}(\sin18^\circ + \cos36^\circ) \]
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