A cos a+ b sin a= m and a sin a- b cos a=n then a(square)+b(square)=?
Answers
Answer:-
Explanation:-
Given:-
a CosA + b SinA = m........(i)
a SinA - b CosA = n..........(ii)
Answer:-
m = a CosA + b SinA = m
Square both side in equation (i)
m² = (a CosA + b SinA)²
m² = a² Cos²A + b Sin²A + 2a CosA * b SinA...........(iii)
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n = a SinA - b CosA
Square both side in equation (ii)
n² = (a SinA - b CosA)²
n² = a² Cos²A + b² Sin²A - 2a CosA * b SinA..........(iv)
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Add equation (iii) and (iv)
m² + n² = a² Sin²A + b² Cos²A + a² Cos²A + b² Sin²A
[ 2a CosA * b SinA and - 2 CosA * b SinA are canceled while addition]
We know that Sin²A + Cos²A = 1
m² + n² = a² + b²
So,the value is a² + b²
→ m² = (a cosA + b sinA)²
→ m² = a² cos²A + b sin²A + 2a cosA· b sinA
→ n² = a² sin²A + b² cos²A - 2 a cos·b sinA
On adding m² and n², 2 acosA·bsina cancel out.
→ m² + n² = a²sin²A + b cos²A + a²cos²A + b² sin²A
→ m² + n² = a²(sin²A + cos²A) + b²(sin²A + cos²A)
Since sin²A + cos²A = 1, hence
→ m² + n² = a² + b²
Hence required answer is m² + n².