The two identities giving the alternative forms for cos 2θ lead to the following equations:

The sign of the square root needs to be chosen properly—note that if π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore the correct sign to use depends on the value of θ.

For the tan function, the equation is:

Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:

Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:

This also gives:

Similar manipulations for the cot function give:

Sine and angle ratio identity
अर्ध कोणों की सर्वसमिकाएँ
Proof of Compositions of trig and inverse trig functions
कोज्या (Cosine)