If (sinθ)/x = (cosθ)/y, prove that sinθ − cosθ = (x − y)/√(x² + y²).
यदि (sinθ)/x = (cosθ)/y हो, तो सिद्ध कीजिए कि sinθ − cosθ = (x − y)/√(x² + y²)।
This Question Asked In:
SSC, CGL MAINS 2026
Correct Answer :B - (x − y)/√(x² + y²)
Detailed Text Solution
Given:
(sinθ)/x = (cosθ)/y
Cross multiplying:
y sinθ = x cosθ
Dividing both sides by cosθ:
y tanθ = x
tanθ = x / y
Using identity:
sinθ = tanθ / √(1 + tan²θ)
= (x/y) / √(1 + x²/y²)
= x / √(x² + y²)
Similarly,
cosθ = y / √(x² + y²)
Therefore,
sinθ − cosθ = x/√(x² + y²) − y/√(x² + y²)
= (x − y)/√(x² + y²).