Quotient Trigonometric identities

In this section we will discuss the quotient trigonometric identities and their proofs.
The quotient identities on trigonometry are as follows :
1) $tan \Theta = \frac{sin \Theta}{cos \Theta}$

2) $cot \Theta = \frac{cos \Theta}{sin \Theta}$

Proof : Let a revolving ray start from OX and revolve into the position OP to trace out any angle $\Theta$ in any of the foru quadrants. From 'P' draw perpendicular to x-axis.

1) In the right angled triangle OMP, we have,
$sin \Theta = \frac{PM}{OP}$ ----------(i)

$cos \Theta = \frac{OM}{OP}$ ----------(ii)

Now dividing equation (i) by (ii)
$\frac{sin \Theta}{cos \Theta} = \frac{\left ( {\frac{PM}{OP}} \right )}{\left ({\frac{OM}{OP}} \right )}$

          = ${\left ( {\frac{PM}{OP}} \right )}\times{\left ({\frac{OP}{OM}} \right )}$

          = $\frac{PM}{OM}$
          = $tan \Theta$
∴ $\frac{sin \Theta}{cos \Theta} = tan \Theta$

2) In the right angled triangle OMP, we have,
$sin \Theta = \frac{PM}{OP}$ ----------(i)

$cos \Theta = \frac{OM}{OP}$ ----------(ii)

Now dividing equation (ii) by (i)
$\frac{cos \Theta}{sin \Theta} = \frac{\left ( {\frac{OM}{OP}} \right )}{\left ({\frac{PM}{OP}} \right )}$

          = ${\left ( {\frac{OM}{OP}} \right )}\times{\left ({\frac{OP}{PM}} \right )}$

          = $\frac{OM}{PM}$
          = $cot \Theta$
∴ $\frac{cos \Theta}{sin \Theta} = cot \Theta$

Quotient trigonometric identities

1) $tan \Theta \times cos \Theta = sin \Theta$
2) $cot \Theta \times sin \Theta = cos \Theta$


11th grade math

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