Write the formulas of trigonometric identities.

Trigonometric identities :

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. The trigonometric identities hold true only for the right-angle triangle.

Reciprocal Identities:

• $Sin θ = \frac{1}{Cosec θ}$ or $Cosec θ = \frac{1}{Sin θ}$
• $Cos θ = \frac{1}{Sec θ}$ or $Sec θ = \frac{1}{Cos θ}$
• $Tan θ = \frac{1}{Cot θ}$ or $Cot θ = \frac{1}{Tan θ}$

Pythagorean Identities:

• $Sin^2 \theta + cos^2 \theta = 1$
• $1 + tan^2 \theta = sec^2 \theta$
• $cosec^2 \theta = 1 + cot^2 \theta $

Ratio Identities:

• $Tan θ = \frac{Sin θ}{Cos θ}$
• $Cot θ = \frac{Cos θ}{Sin θ}$

Opposite Angle Identities:

• $Sin (-θ) = – Sin θ$
• $Cos (-θ) = Cos θ$
• $Tan (-θ) = – Tan θ$
• $Cot (-θ) = – Cot θ$
• $Sec (-θ) = Sec θ$
• $Cosec (-θ) = -Cosec θ$

Complementary Angles Identities:

• $Sin (90 – θ) = Cos θ$
• $Cos (90 – θ) = Sin θ$
• $Tan (90 – θ) = Cot θ$
• $Cot ( 90 – θ) = Tan θ$
• $Sec (90 – θ) = Cosec θ$
• $Cosec (90 – θ) = Sec θ$

Angle Sum and Difference Identities:

Consider two angles, A and B, the trigonometric sum and difference identities are as follows:

• $sin(A+B)=sin(A).cos(B)+cos(A).sin(B)$
• $sin(A–B)=sinA.cosB–cosA.sinB$
• $cos(A+B)=cosA.cosB–sinA.sinB$
• $cos(A–B)=cosA.cosB+sinA.sinB$

• $tan (A + B) = \frac{tan A + tan B}{1 - tan A.tan B}$
• $tan (A - B) = \frac{tan A - tan B}{1 + tan A.tan B}$