A basic mathematical relation between any trigonometric functions is called a basic trigonometric identity. A basic trigonometric identity is usually used as a formula in mathematics. Hence, it is also called as a basic trigonometric formula.
There are four types of basic trigonometric identities in trigonometry and they are used as formulas in mathematics. So, everyone who studies the trigonometry newly must firstly learn all of these basic trigonometric identities.
The following trigonometric formulas derived by taking theta ($\theta$) as angle of a right angled triangle.
Trigonometric ratios form six identities in reciprocal form and learn proofs of these reciprocal formulas.
$(1)\,\,\,\,$ $\sin \theta \,=\, \dfrac{1}{\csc \theta}$
$(2)\,\,\,\,$ $\cos \theta \,=\, \dfrac{1}{\sec \theta}$
$(3)\,\,\,\,$ $\tan \theta \,=\, \dfrac{1}{\cot \theta}$
$(4)\,\,\,\,$ $\cot \theta \,=\, \dfrac{1}{\tan \theta}$
$(5)\,\,\,\,$ $\sec \theta \,=\, \dfrac{1}{\cos \theta}$
$(6)\,\,\,\,$ $\csc \theta \,=\, \dfrac{1}{\sin \theta}$
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 $
The six trigonometric functions involve in two relations in quotient form and learn the proofs of quotient identities.
$(1)\,\,\,\,$ $\dfrac{\sin \theta}{\cos \theta} = \tan \theta$
$(2)\,\,\,\,$ $\dfrac{\cos \theta}{\sin \theta} = \cot \theta$
The six trigonometric functions form three Pythagorean identities on the basis of Pythagoras Theorem.
$(1)\,\,\,\,$ $\sin^2{\theta} \,+\, \cos^2{\theta} \,=\, 1$
$(2)\,\,\,\,$ $\sec^2{\theta} \,-\, \tan^2{\theta} \,=\, 1$
$(3)\,\,\,\,$ $\csc^2{\theta} \,-\, \cot^2{\theta} \,=\, 1$
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