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Let us recall that if the sum of the two acute angles is \(90^{\circ}\), then the angles are said to be complementary.
 
In a right-angled triangle, the sum of two acute angles are \(90^{\circ}\).
 
That is, we can say that the two acute angles in a right-angled triangle are complementary.
 
Consider the triangle \(PQR\) right-angled at \(P\).
 
complementarty angles.png
 
Here \(R\) and \(Q\) are complementary angles.
 
Therefore if \(\angle R\) \(=\) \(\theta\), then \(\angle Q\) \(=\) \(90^{\circ} - \theta\).
 
Let us write all the trigonometric ratios with respect to \(\angle R\) \(=\) \(\theta\) in a table.
 
Table \(1\):
 
Trigonometric ratio
Relationship with the \(\Delta PQR\)
Trigonometric ratio
Relationship with the \(\Delta PQR\)
\(\sin \theta\)
\(\sin \theta\) \(=\) \(\frac{PQ}{RQ}\)
\(\text{cosec}\,\theta\)
\(\text{cosec}\,\theta\) \(=\) \(\frac{RQ}{PQ}\)
\(\cos \theta\)
\(\cos \theta\) \(=\) \(\frac{PR}{RQ}\)
\(\sec \theta\)
\(\sec \theta\) \(=\) \(\frac{RQ}{PR}\)
\(\tan \theta\)
\(\tan \theta\) \(=\) \(\frac{PQ}{PR}\)
\(\cot \theta\)
\(\cot \theta\) \(=\) \(\frac{PR}{PQ}\)
 
Now let us write all the trigonometric ratios with respect to \(\angle Q\) \(=\) \(90^{\circ} - \theta\) in a table.
 
Table \(2\):
 
Trigonometric ratio
Relationship with the \(\Delta PQR\)
Trigonometric ratio
Relationship with the \(\Delta PQR\)
\(\sin (90^{\circ} - \theta)\)
\(\sin (90^{\circ} - \theta)\) \(=\) \(\frac{PR}{RQ}\)
\(\text{cosec}\,(90^{\circ} - \theta)\)
\(\text{cosec}\,(90^{\circ} - \theta)\) \(=\) \(\frac{RQ}{PR}\)
\(\cos (90^{\circ} - \theta)\)
\(\cos (90^{\circ} - \theta)\) \(=\) \(\frac{PQ}{RQ}\)
\(\sec (90^{\circ} - \theta)\)
\(\sec (90^{\circ} - \theta)\) \(=\) \(\frac{RQ}{PQ}\)
\(\tan (90^{\circ} - \theta)\)
\(\tan (90^{\circ} - \theta)\) \(=\) \(\frac{PR}{PQ}\)
\(\cot (90^{\circ} - \theta)\)
\(\cot (90^{\circ} - \theta)\) \(=\) \(\frac{PQ}{PR}\)
 
Comparing the tables \(1\) and \(2\), we arrive at the following identities.
 
1. \(\sin \theta\) \(=\) \(\cos (90^{\circ} - \theta)\)
 
2. \(\cos \theta\) \(=\) \(\sin (90^{\circ} - \theta)\)
 
3. \(\tan \theta\) \(=\) \(\cot (90^{\circ} - \theta)\)
 
4. \(\text{cosec}\,\theta\) \(=\) \(\sec (90^{\circ} - \theta)\)
 
5. \(\sec \theta\) \(=\) \(\text{cosec}(90^{\circ} - \theta)\)
 
6. \(\cot \theta\) \(=\) \(\tan (90^{\circ} - \theta)\)