Trigonometric functions and their properties

(\sec \theta) is the reciprocal of (\cos \theta) and remains positive in the fourth quadrant. So (\sec(2\pi-\theta)=\sec \theta).

(\cosec \theta) is the reciprocal of (\sin \theta) and is negative in the fourth quadrant. Therefore the value is (-\cosec \theta).

The value of (\sin \theta) lies in ([-1,1]), so the maximum value of (\sin^2 \theta) is (1). Squaring makes the value non-negative.

For every real \(\theta\), \(-1 \leq \cos\theta \leq 1\). Hence, \(\cos^2\theta\) lies in the interval from \(0\) to \(1\). When \(\cos\theta=0\), for example at \(\theta=\frac{\pi}{2}\), we get \(\cos^2\theta=0\). Therefore, the minimum value is \(0\). The value \(1\) is the maximum, not the minimum. Exam tip: the square of a real number cannot be negative.

(\sin^2 \theta+\cos^2 \theta=1), so it is a constant value. Identify the basic identity directly.

Putting (\sin \theta=0) in (\sin^2 \theta+\cos^2 \theta=1) gives (\cos^2 \theta=1). Use the identity in such questions.

Putting (\cos \theta=0) in (\sin^2 \theta+\cos^2 \theta=1) gives (\sin^2 \theta=1). Substitute the known value in the formula.

(\1+\tan^2 \theta=\sec^2 \theta) and (\frac{1}{\sec^2 \theta}=\cos^2 \theta). Be careful while taking reciprocals.

(\1+\cot^2 \theta=\cosec^2 \theta) and (\frac{1}{\cosec^2 \theta}=\sin^2 \theta). Simplify fractions using identities.

From (\sec^2 \theta=1+\tan^2 \theta), we get (\sec^2 \theta-1=\tan^2 \theta). Rearrange the formula into a simple form.

From (\cosec^2 \theta=1+\cot^2 \theta), (\cosec^2 \theta-1=\cot^2 \theta). Practice similar identities in pairs.

Since (\cosec \theta=\frac{1}{\sin \theta}), (\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta). Convert reciprocal relations into fractions.

Since (\sec \theta=\frac{1}{\cos \theta}), (\frac{\cos \theta}{\sec \theta}=\cos^2 \theta). Writing the reciprocal makes simplification easy.

(\sin \theta) and (\cosec \theta) are reciprocals, so their product is (1). Remember reciprocal pairs.

\(\cos \theta\) and \(\sec \theta\) are reciprocals of each other. Therefore their product is (1).

Using the definitions, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). Therefore, \(\tan\theta\cdot\cot\theta=\frac{\sin\theta}{\cos\theta}\cdot\frac{\cos\theta}{\sin\theta}=1\), wherever both functions are defined. It is not \(0\); tangent and cotangent are reciprocal ratios. Exam tip: the product of reciprocal trigonometric ratios is usually \(1\).

Using (\tan \theta=\frac{\sin \theta}{\cos \theta}) and (\sec \theta=\frac{1}{\cos \theta}), the answer is (\sin \theta). Convert division into fractions and cancel.

Using (\cot \theta=\frac{\cos \theta}{\sin \theta}) and (\cosec \theta=\frac{1}{\sin \theta}), we get (\cos \theta). Writing standard forms is the safest method.

From (\sec \theta=\frac{1}{\cos \theta}) and (\tan \theta=\frac{\sin \theta}{\cos \theta}), we get (\frac{1}{\sin \theta}=\cosec \theta). Simplify fractions carefully.

Using \(\cosec\theta=\frac{1}{\sin\theta}\) and \(\cot\theta=\frac{\cos\theta}{\sin\theta}\),
\(\frac{\cosec\theta}{\cot\theta}=\frac{1/\sin\theta}{\cos\theta/\sin\theta}=\frac{1}{\cos\theta}=\sec\theta\). Hence, \(\sec\theta\) is correct. \(\tan\theta\) equals \(\frac{\sin\theta}{\cos\theta}\), so it is not the answer. Exam tip: rewrite reciprocal and quotient trigonometric ratios in terms of \(\sin\theta\) and \(\cos\theta\) before simplifying.