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Equations Involving sinx + cosx and sinx.cosx

Express each ...

Question

Express each of the following as the sum or difference of sines and cosines :
(i) $2 s i n 3 θ c o s θ$
(ii) $2 c o s 3 θ s i n 2 θ$
(iii) $2 s i n 4 θ s i n 3 θ$
(iv) $2 c o s 7 θ c o s 3 θ$

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Solution

(i) $2 s i n 3 θ c o s θ = s i n (3 θ + θ) + s i n (3 θ - θ) [∵ 2 s i n A c o s B = s i n (A + B) + s i n (A - B)] = s i n 4 θ + s i n 2 θ$

(ii) $2 c o s 3 θ s i n 2 θ ∵ 2 c o s A s i n B = s i n (A + B) - s i n (A - B) \Rightarrow 2 c o s 3 θ s i n 2 θ = s i n (3 θ + 2 θ) - s i n (3 θ - 2 θ) = s i n 5 θ - s i n θ$

(iii) $2 s i n 4 θ s i n 3 θ ∵ 2 s i n A s i n B = c o s (A - B) - c o s (A + B) \Rightarrow 2 s i n 4 θ s i n 3 θ = c o s (4 θ - 3 θ) - c o s (4 θ + 3 θ) = c o s θ - c o s 7 θ$

(iv) $2 c o s 7 θ c o s 3 θ ∵ 2 c o s A c o s B = c o s (A + B) + c o s (A - B) \Rightarrow 2 c o s 7 θ c o s 3 θ = c o s (7 θ + 3 θ) + c o s (7 θ - 3 θ) = c o s 10 θ + c o s 4 θ$

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Equations Involving sinx + cosx and sinx.cosx

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