The value of $\sin 21°\cos 9°-\cos 84°\cos 6°$ is

Step 1: Simplify sin(21°) × cos(9°) using the product-to-sum identity:

sin(A) × cos(B) = ½ [sin(A + B) + sin(A − B)]

Let A = 21° and B = 9°:

sin(21°) × cos(9°) = ½ [sin(21° + 9°) + sin(21° − 9°)]

= ½ [sin(30°) + sin(12°)]

Step 2: Simplify cos(84°) × cos(6°) using the product-to-sum identity:

cos(A) × cos(B) = ½ [cos(A + B) + cos(A − B)]

Let A = 84° and B = 6°:

cos(84°) × cos(6°) = ½ [cos(84° + 6°) + cos(84° − 6°)]

= ½ [cos(90°) + cos(78°)]

Since cos(90°) = 0:

= ½ [0 + cos(78°)]

= ½ cos(78°)

Step 3: Combine the results:

sin(21°) × cos(9°) − cos(84°) × cos(6°)

= ½ [sin(30°) + sin(12°)] − ½ cos(78°)

Since cos(78°) = sin(12°):

= ½ [sin(30°) + sin(12°) − sin(12°)]

= ½ sin(30°)

Since sin(30°) = ½:

= ½ × ½

= ¼