Express sin5A in terms of cosA.

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Solution

Dear student
$\sin 5 A = \sin (3 A + 2 A) = S i n 3 A \cos 2 A + \cos 3 A \sin 2 A = (3 \sin A - 4 \sin^{3} A) (2 \cos^{2} A - 1) + (4 \cos^{3} A - 3 \cos A) \times 2 s o n A \cos A = - 3 \sin A + 4 \sin^{3} A + 6 \sin A \cos^{2} A - 8 \sin^{3} A \cos^{2} A + 8 \sin A \cos^{4} A - 6 \sin A \cos^{2} A = 8 \sin A \cos^{4} A - 8 \sin^{3} A \cos^{2} A - 2 \sin A + 4 \sin^{3} A = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - 3 \sin A + 3 \sin A \cos^{4} A + 4 \sin^{3} A + 2 \sin^{3} A \cos^{2} A = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - 3 s i n A (1 - \cos^{4} A) + 2 \sin^{3} A (2 + \cos^{2} A) = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - 3 s i n A (1 - c o s^{2} A) (1 + c o s^{2} A) + 2 s i n^{3} A (2 + c o s^{2} A) = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - 3 s i n^{3} A (1 + c o s^{2} A) + 2 s i n^{3} A (2 + c o s^{2} A) = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [3 (1 + \cos^{2} A)] + 2 s i n^{3} A (2 + c o s^{2} A) = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [3 + 3 \cos^{2} A - 4 - 2 \cos^{2} A] = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [\cos^{2} A - 1] = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A (- \sin^{2} A) = 5 s i n A c o s^{4} A - 10 s i n^{3} A c o s^{2} A + s i n^{5} A Y o u c a n l e a v e t h i s h e r e b e c a u s e i t c a' t b e f u l l y e x p r e s s e d i n t e r m s o f C o s A$
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