First slide

Multiple and sub- multiple Angles

Question

If $\sin â¡ (Î¸ / 2) = a, \cos â¡ (Î¸ / 2) = b$ , then $(1 + \sin â¡ Î¸) (3 \sin â¡ Î¸ + 4 \cos â¡ Î¸ + 5) =$

Moderate

A

$(a + b)^{2} (a + 3 b)^{2}$
B

$(a + b)^{2} (3 a + b)^{2}$
C

$(a â b)^{2} (a â 3 b)^{2}$
D

$(a â b)^{2} (3 a â b)^{2}$

Solution

$\begin{array}{l} (1 + \sin â¡ Î¸) (3 \sin â¡ Î¸ + 4 \cos â¡ Î¸ + 5) \\ = (\cos â¡ (Î¸ / 2) + \sin â¡ (Î¸ / 2))^{2} [6 \sin â¡ (Î¸ / 2) \cos â¡ (Î¸ / 2) + 4 (2 \cos^{2} â¡ Î¸ / 2 â 1) + 5] \\ = (a + b)^{2} [6 a b + 8 b^{2} + a^{2} + b^{2}] \\ = (a + b)^{2} (a + 3 b)^{2} \end{array}$

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