Math  /  Trigonometry

Questionb) cos10cos7sin10sin7\cos 10^{\circ} \cos 7^{\circ}-\sin 10^{\circ} \sin 7^{\circ} cos(10+7)2cos\cos (10+7)-2 \cos d) sinπ3cosπ4cosπ3sinπ4\sin \frac{\pi}{3} \cos \frac{\pi}{4}-\cos \frac{\pi}{3} \sin \frac{\pi}{4}

Studdy Solution
Recognize the trigonometric identity for the expression in part d:
The expression sinπ3cosπ4cosπ3sinπ4\sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4} matches the sine of a difference identity:
sinAcosBcosAsinB=sin(AB)\sin A \cos B - \cos A \sin B = \sin(A - B)
Apply the identity:
sin(π3π4)\sin\left(\frac{\pi}{3} - \frac{\pi}{4}\right)
Calculate the angle:
π3π4=4π123π12=π12\frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12}
So, the expression simplifies to:
sin(π12)\boxed{\sin\left(\frac{\pi}{12}\right)}

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