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

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$\frac{1}{8}$

$\frac{3}{8}$

$\frac{1}{4}$

$\frac{3}{2}$

answer is A.

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Detailed Solution

$\sin 21 ° \cos 9 ° - \cos 84 ° \cos 6 °$

$= \frac{1}{2} [2 \sin 21 ° \cos 9 ° - 2 \cos 84 ° \cos 6 °]$

$= \frac{1}{2} [\sin (21 ° + 9 °) + \sin (21 ° - 9 °) - \cos (84 ° + 6 °) - \cos (84 ° - 6 °)]$

$= \frac{1}{2} [\sin 30 ° + \sin 12 ° - \cos 90 ° - \cos 78 °]$

$= \frac{1}{2} [\frac{1}{2} + \sin 12 ° - 0 - \cos (90 ° - 12 °)]$

$= \frac{1}{2} [\frac{1}{2} + \sin 12 ° - \sin 12 °]$

$= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Explanation:

Expression:

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

Step-by-Step Solution:

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

Final Answer

The value of the expression is ¼.

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