If x\sin^{3}\theta+y\cos^{3}\theta=\sin \theta \cos \theta and x\sin \theta-y\cos \theta=0, for every \theta\in(0,\frac{\pi}{2}) then what is x^{2}+y^{2} equal to?

Explanation

From x\sin\theta = y\cos\theta, we get x = y\frac{\cos\theta}{\sin\theta}. Substitute this into the first equation: (y\frac{\cos\theta}{\sin\theta})\sin^3\theta + y\cos^3\theta = \sin\theta\cos\theta \implies y\cos\theta\sin^2\theta + y\cos^3\theta = \sin\theta\cos\theta \implies y\cos\theta(\sin^2\theta+\cos^2\theta) = \sin\theta\cos\theta. Thus, y\cos\theta = \sin\theta\cos\theta \implies y = \sin\theta. Consequently, x = \cos\theta. Therefore, x^2+y^2 = \cos^2\theta+\sin^2\theta = 1.

Related questions on Trigonometry

• Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?
• If \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4, then which one of the following is a value of $(\tan^2 \...
• For 0 < \theta < \frac{\pi}{2}, consider the following : I. $(\tan^4 \theta + \tan^6 \theta)(\cot^4 \theta + \cot^6 \theta) = \sec^2 \the...
• If 3\sin \theta + 4\cos \theta = 5, then what is a value of 4\tan \theta + 3\cot \theta ?
• At a point on level ground, the tangent of the angle of elevation of the top of a tower is found to be \frac{5}{6}. On walking 70 m toward...