Important Theorems and Formulas Developed by Indian Mathematicians

Aryabhata, a renowned Indian mathematician and astronomer, is widely recognized for his discovery of the sine rule, which establishes a relationship between the sides and angles of a triangle.

The formula $\sin^2 \theta + \cos^2 \theta = 1$ is a fundamental trigonometric identity that expresses the relationship between the sine and cosine functions of an angle.

Bhaskara II, a prominent Indian mathematician and astronomer, is credited with developing the concept of half-angle formulas in trigonometry, which provide expressions for sine, cosine, and tangent of half an angle.

The formula $\sin 2\theta = 2 \sin \theta \cos \theta$ is a double-angle formula in trigonometry, which expresses the sine of twice an angle in terms of the sine and cosine of the original angle.

Madhava of Sangamagrama, a renowned Indian mathematician and astronomer, made significant contributions to the expansion of trigonometric functions into infinite series, laying the groundwork for the development of calculus.

The formula $\cos \theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}$ is a trigonometric identity that expresses the relationship between the cosine and tangent functions of an angle.

Aryabhata, a pioneering Indian mathematician and astronomer, developed the concept of the sine and cosine tables, which provided pre-calculated values of trigonometric ratios for various angles, simplifying trigonometric calculations.

The formula $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is a fundamental trigonometric identity that expresses the relationship between the tangent, sine, and cosine functions of an angle.

Brahmagupta, a renowned Indian mathematician and astronomer, introduced the concept of the versine function, which is defined as $1 - \cos \theta$ and is used in various trigonometric calculations.

The formula $\sin (A + B) = \sin A \cos B + \cos A \sin B$ is a sum-to-product formula in trigonometry, which expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles.

Madhava of Sangamagrama, a prominent Indian mathematician and astronomer, introduced the concept of the haversine function, which is defined as $\frac{1 - \cos \theta}{2}$ and is used in various trigonometric calculations, including the calculation of great-circle distances.

The formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$ is a sum-to-product formula in trigonometry, which expresses the cosine of the sum of two angles in terms of the cosines and sines of the individual angles.

Bhaskara II, a renowned Indian mathematician and astronomer, introduced the concept of the cotangent function, which is defined as the ratio of the cosine to the sine of an angle.

The formula $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ is a sum-to-product formula in trigonometry, which expresses the tangent of the sum of two angles in terms of the tangents of the individual angles.

Bhaskara II, a prominent Indian mathematician and astronomer, introduced the concepts of the secant and cosecant functions, which are defined as the reciprocals of the cosine and sine functions, respectively.