If $x = a s i n A c o s C, y = a s i n A s i n C, z = a c o s A$ shwo that $a^{2} = x^{2} + y^{2} + z^{2}$ [4 MARKS]

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Solution

Formula: 2 Marks
Proof: 2 Marks

RHS $= x^{2} + y^{2} + z^{2} = (a s i n A c o s C)^{2} + (a s i n A s i n C)^{2} + (a c o s A)^{2} = a^{2} s i n^{2} A c o s^{2} C + a^{2} s i n^{2} A s i n^{2} C + a^{2} c o s^{2} A = a^{2} [s i n^{2} A c o s^{2} C + s i n^{2} A s i n^{2} C + c o s^{2} A] = a^{2} [s i n^{2} A (c o s^{2} C + s i n^{2} C) + c o s^{2} A] = a^{2} (s i n^{2} A + c o s^{2} A) [∵ c o s^{2} C + s i n^{2} C = 1] = a^{2} = L H S$

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