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expres the following as sums and differences of sines or cosines

cos8t * sin2t

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

replacing by by -b and using that
cos(-b)= cos(b)
sin(-b)= -sin(b)

gives:

sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

Add the two equations:

sin(a+b) + sin(a-b) = 2sin(a)cos(b)-->

sin(a)cos(b)= 1/2 [sin(a+b) + sin(a-b)]

who found trigonometry?

Trigonometry was not discovered by a single person. Its development spans thousands of years and involves contributions from various civilizations, including the Babylonians, Egyptians, Greeks, and Indians. Some notable mathematicians who contributed to the development of trigonometry include Hipparchus, Menelaus, Ptolemy, Aryabhata, and Al-Khwarizmi.

Trigonometry is a branch of mathematics that has been developed and studied by different civilizations throughout history. Its origins can be traced back to ancient civilizations such as the Babylonians, Egyptians, and Greeks. The Greek mathematician Hipparchus is often credited with the development of the basic principles of trigonometry as we know it today. However, many other mathematicians and scholars from various cultures have contributed to the development and advancement of trigonometry over centuries.

Trigonometry is a branch of mathematics that has been developed and used by various civilizations throughout history. The ancient Egyptians, Babylonians, and Greeks all made significant contributions to trigonometry. However, the systematic study and formalization of trigonometry can be attributed to the ancient Greeks, particularly the mathematicians Hipparchus and Ptolemy. They developed trigonometry as a means to understand and calculate angles and distances in geometric figures and astronomical observations. Over time, trigonometry has continued to evolve and expand through the work of many mathematicians and scholars from different cultures. It has become an essential tool in various fields, including physics, engineering, and navigation.