The basic relation between any two or more trigonometric functions is called basic trigonometric identity or basic trigonometric formula.

In trigonometry, the six trigonometric ratios form four types of basic trigonometric formulas. Here is the list of four types of basic trigonometric identities, which are derived by assuming theta ($\theta$) as the angle of the right angled triangle.

1

Trigonometric ratios form six identities in reciprocal form and learn proofs of these reciprocal formulas.

$(1)\,\,\,\,$ $\sin \theta \,=\, \dfrac{1}{\csc \theta}$

$(2)\,\,\,\,$ $\cos \theta \,=\, \dfrac{1}{\sec \theta}$

$(3)\,\,\,\,$ $\tan \theta \,=\, \dfrac{1}{\cot \theta}$

$(4)\,\,\,\,$ $\cot \theta \,=\, \dfrac{1}{\tan \theta}$

$(5)\,\,\,\,$ $\sec \theta \,=\, \dfrac{1}{\cos \theta}$

$(6)\,\,\,\,$ $\csc \theta \,=\, \dfrac{1}{\sin \theta}$

2

Trigonometric functions form three formulas in product form and learn the proofs of product identities.

$(1)\,\,\,\,$ $\sin \theta \times \csc \theta = 1 $

$(2)\,\,\,\,$ $\cos \theta \times \sec \theta = 1 $

$(3)\,\,\,\,$ $\tan \theta \times \cot \theta = 1 $

3

The six trigonometric functions involve in two relations in quotient form and learn the proofs of quotient identities.

$(1)\,\,\,\,$ $\dfrac{\sin \theta}{\cos \theta} = \tan \theta$

$(2)\,\,\,\,$ $\dfrac{\cos \theta}{\sin \theta} = \cot \theta$

4

The six trigonometric ratios form three formulas, which are derived by the Pythagoras theorem and learn proofs of Pythagorean formulas.

$(1)\,\,\,\,$ $\sin^2 \theta \,+\, \cos^2 \theta = 1$

$(2)\,\,\,\,$ $\sec^2 \theta \,-\, \tan^2 \theta = 1$

$(3)\,\,\,\,$ $\csc^2 \theta \,-\, \cot^2 \theta = 1$

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