S L Loney Solutions for Chapter: General Expressions for All Angles Having a Given Trigonometrical Ratio, Exercise 1: EXAMPLES XI

What are the most general values of $\theta$ which satisfy the following equation: $4{\sin }^2\theta =3$?

What are the most general values of $\theta$ which satisfy the following equation: $2{\cot }^2\theta ={\mathrm{cosec}}^2\theta$?

What are the most general values of $\theta$ which satisfy the following equation: ${\sec }^2\theta =\frac{4}{3}$?

What is the most general value of $\theta$ that satisfies both of the equations $\cos \theta =-\frac{1}{\sqrt{2}} \& \tan \theta =1$?

What is the most general value of $\theta$ that satisfies both of the equations: $\cot \theta =-\sqrt{3} \& \mathrm{cosec}\theta =-2$?

If $\theta -\alpha =n\pi +(-1{)}^n\beta$ prove that $\theta =2m\pi +\alpha +\beta$ or else that $\theta =\left(2m+1\right)\pi +\alpha -\beta$, where $m \& n$ are any integers. 

If $\cos p\theta +\cos q\theta =0,$ prove that the different value of $\theta$ form two arithmetical progression in which the common differences are  $\frac{2\pi }{p+q}$ and $\frac{2\pi }{p-q}$ respectively. 

Construct the angle whose sine is $\frac{3}{2+\sqrt{5}}$.