Trigonometric functions and their properties

The basic identity is (1+\tan^2 x=\sec^2 x). In exams, it can also be derived from (\sin^2 x+\cos^2 x=1).

Use the identity \(\cosec^2 x=1+\cot^2 x\). Given \(\cot x=3\), we get \(\cot^2 x=9\). Therefore, \(\cosec^2 x=1+9=10\). Option 9 is only the value of \(\cot^2 x\), not of \(\cosec^2 x\). Exam tip: In the identity \(1+\cot^2 x=\cosec^2 x\), do not forget to add 1.

This is the form of (\sin(75^\circ-15^\circ)). Hence the value is (\sin 60^\circ=\frac{\sqrt{3}}{2}).

Since \(\tan x=\frac{\sin x}{\cos x}\), it is undefined wherever \(\cos x=0\). This occurs at \(x=\frac{(2n+1)\pi}{2}\). Option C excludes zeros of sine, which affect cotangent instead. Exam tip: check the denominator first.

When \(\sin x=\sin y\), the two general families are \(x=2n\pi+y\) and \(x=(2n+1)\pi-y\), where \(n\in\mathbb Z\). The first follows from periodicity, while the second uses \(\sin(\pi-y)=\sin y\). Option B represents only some cases and misses the second family. Exam tip: for equal sine values, remember the angles \(y\) and \(\pi-y\).

For (\cos x=\cos y), angles are of the form (2n\pi\pm y). In exams, remember the even nature of cosine.

Since \(\cos(-2x)=\cos 2x\), \(\cos 2x\) is even. Replacing \(x\) by \(x+\pi\) leaves it unchanged, and its range is \([-1,1]\). Although \(\sec 2x\) is even, its range excludes values between \(-1\) and \(1\). Exam tip: test \(f(-x)\) for parity.

Using the formulas for \(\tan\left(\frac{\pi}{4}+x\right)\) and \(\tan\left(\frac{\pi}{4}-x\right)\), the product is (1). This is a useful standard result.

In a right triangle, perpendicular is (7), base is (24), and hypotenuse is (25). Hence (\sin x=\frac{7}{25}).

The period of (\cos x) is (2\pi) and that of (\cos 2x) is (\pi). Their common fundamental period is (2\pi).

For any real (x), (-1\leq \sin x\leq 1). In exams, apply the basic range of sine and cosine first.

Since (\tan x=\frac{\sin x}{\cos x}), it is undefined when (\cos x=0). Thus (x=\frac{\pi}{2}+n\pi).

Since \(\cosec x=\frac{1}{\sin x}\), it is undefined when \(\sin x=0\). Thus \(x=n\pi\).

We use \(\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)\). The maximum \(\sqrt{2}\) occurs at \(x=\frac{\pi}{4}\).

By the double angle formula, (2\sin x\cos x=\sin 2x). This identity is used frequently in exams.

The double angle identity is (\cos 2x=\cos^2 x-\sin^2 x). In exams, remember all three forms of (\cos 2x).

Since \(\sec x=1/\cos x\), it is undefined when \(\cos x=0\), i.e. at \(x=\frac{(2n+1)\pi}{2}\). Also, \(|\sec x|\ge1\), giving the stated range. For \(\csc x\), the excluded values are \(n\pi\). Exam tip: first locate angles that make the denominator zero in reciprocal functions.

We have (\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}). Simplifying gives (\frac{1}{\sin x\cos x}=\sec x\cosec x).

The period of (\tan ax) is ( \frac{\pi}{|a|} ). Here (a=3), so the period is ( \frac{\pi}{3} ).

The period of (\cos ax) is ( \frac{2\pi}{|a|} ). Hence the period of (\cos 4x) is ( \frac{\pi}{2} ).