Answer to Question #198417 in Trigonometry for Steven

Question

Solve (sinx)sec^2(x)+(cos x)cosec^2 (x)=0.

Answer 1

Calculation.

$\qquad\qquad \begin{aligned} \small \sin\theta.\sec^2\theta+\cos\theta.\csc^2\theta&=\small 0\\ \small \frac{\sin\theta}{\cos^2\theta}+\frac{\cos\theta}{\sin^2\theta}&=\small 0\\ \small \frac{\sin^3\theta+\cos^3\theta}{\sin^2\theta.\cos^2\theta}&=\small0\\ \\ \small \sin^2\theta.\cos^2\theta&\not=\small 0\\ \\ \therefore\,\small \sin^3\theta+\cos^3\theta&=\small 0\\ \small (\sin\theta+\cos\theta)(1-\sin\theta.\cos\theta)&=\small 0\\ \\ \small 1-\sin\theta.\cos\theta&=\small 0\\ \small \sin2\theta&=\small 2\\ \because \,\small -1\leq&\small \sin2\theta\leq1\\ \therefore\,\text{neglected},\\ \\ \small \sin\theta+\cos\theta&=\small 0\\ \small \tan\theta&=\small -1\cdots(\cos\theta\not=0)\\ \small \theta&=\small \frac{3\pi}{4}+n\pi\cdots\cdots(\text{Answer})\\ \end{aligned}$