Question

a cos thita+b sin thita =m ,and a sin thita +b cos thita =n ,then show that : a^2 +b^2 =m^2 +n^2​

Answer 1

Answer:

Given,

a cosΦ + b sinΦ = m

a SinΦ + b cosΦ = n

To prove :- a² + b² = m² + n²

Squaring both sides we get,

a²cos²Φ + b²sin²Φ = m² ••••••• 1

a²sin²Φ + b²cos²Φ = n² ••••••• 2

Adding 1 & 2

m² + n² = a²cos²Φ + a²sin²Φ +

b²sin²Φ + b²cos²Φ

Taking a² and b² common in RHS we get,

m² + n² = a² ( sin²Φ + cos²Φ ) +

b² ( sin²Φ + cos²Φ )

m² + n² = a²(1) + b²(1)

[ Sin²Φ + Cos²Φ = 1 ]

Hence, a² + b² = m² + n²

LHS = RHS ( Hence proved )

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