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प्रश्न
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
उत्तर
Given 3 sin θ + 5 cos θ = 5
Squaring on both sides for both the equations
⇒ 9 sin2θ + 25 cos2θ + 30 sinθ cosθ = 25
⇒ 25 sin2θ + 9 cos2θ − 30 sinθ cosθ = x2
Adding the equations;
⇒ 34 (sin2θ + cos2θ) = 25 + x2
⇒ x2 = 34 − 25 = 9
⇒ x = ±3
∴ 5 sinθ − 3 cosθ = ±3
Hence proved.
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∴ cotθ + tanθ = cosecθ × secθ
