Sum and Difference Trigonometric Formulas
Which of the following is equal to the above expression?
Rectangle \(ABCD\) has side lengths \[\overline{AB}=a, \overline{BC}=24.\] If point \(P\) on \(BC\) internally divides it \(2:1\) and \(\tan (\angle APD)=3\), what is the value of \(a\)?
If \(\sin x=-\frac{3}{11}\), the value of \[\frac{1-\cot x}{1+\tan x}+\frac{1+\cot x}{1-\tan x}\] can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are corpime positive integers. What is \(a+b\)?
If \(\tan \theta + \cot \theta = 16\), what is the value of \(\frac{\csc \theta}{\sec \theta - \tan \theta} + \frac{\csc \theta}{\sec \theta + \tan \theta}\)?
Details and assumptions
Clarification: The function csc, is also known as cosec.
If \(\sin \theta+\cos \theta=\frac{1}{6}\), the value of \[\frac{\csc \theta}{\sec \theta-\tan \theta}+\frac{\csc \theta}{\sec \theta+\tan \theta}\] can be expressed as \(-\frac{a}{b}\), where \(a\) and \(b\) are copime positive integers. What is \(a+b\)?