Reciprocal Identities

A step-by-step guide to reciprocal identities

The reciprocals of the six basic trigonometric functions (\(sin\), \(cos\), \(tan\), \(sec\), \(csc\), \(cot\)) are called reciprocal identities. Reciprocal identities are important trigonometric identities that are used to solve various problems in trigonometry.

The \(sin\) function is the reciprocal of the \(csc\) function and vice-versa; the \(cos\) function is the reciprocal of the \(sec\) function and vice-versa; the \(cot\) function is the reciprocal of the \(tan\) function and vice-versa.

The formulas of the six main reciprocal identities are:

• \(\color{blue}{sin\:\left(\theta \right)=\frac{1}{csc\:\left(\theta \right)}}\)
• \(\color{blue}{cos\:\left(\theta \right)=\frac{1}{sec\:\left(\theta \right)}}\)
• \(\color{blue}{tan\:\left(\theta \right)=\frac{1}{cot\:\left(\theta \right)}}\)
• \(\color{blue}{csc\:\left(\theta \right)=\frac{1}{sin\:\left(\theta \right)}}\)
• \(\color{blue}{sec\:\left(\theta \right)=\frac{1}{cos\:\left(\theta \right)}}\)
• \(\color{blue}{cot\:\left(\theta \right)=\frac{1}{tan\:\left(\theta \right)}}\)

Reciprocal Identities – Example 1:

Find the value of \(sec\: x\) if \(cos\: x = \frac{2}{9}\) using the reciprocal identity.

Solution: 

We know the reciprocal identity \(sec\: x = \frac{1}{cos x}\)

So, if \(cos\: x = \frac{2}{9}\), then:

\(sec\:x=\:\frac{1}{cos\:x}=\frac{1}{\frac{2}{9}}=\frac{9}{2}\)