From the angle sum identities, we get

and

The Pythagorean identities give the two alternative forms for the latter of these:

The angle sum identities also give

It can also be proved using Euler's formula

Squaring both sides yields

But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields

It follows that

Expanding the square and simplifying on the left hand side of the equation gives

Because the imaginary and real parts have to be the same, we are left with the original identities

and also

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ज्या (Sine)
पूरक कोणों से सम्बन्धित सर्वसमिकाएँ