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Introduction - Trigonometric Identities

An identity is an equation that is true for all real values in the domain of the variable.

Based on the above definition, we define a "Trigonometric Identity".

Trigonometric Identity:
A trigonometric identity is an equation involving trigonometric functions that holds true for all angles.

Verifying identities and solving equations depends on the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions.


Example : 1. sinθ . cosec θ = 1

The above is an example of an identity as we know that sinθ and cosecθ are reciprocal to each other and whose product is 1. Similarly following are also examples of an identity.
2. cos θ . sec θ = 1
3. cot
θ . tan θ = 1


Verifying trigonometric identities
Following are the guidelines for verifying the trigonometric identities:

  1. Work with one side of the equation. It is always better to work with the complicated side to start with.
  2. Look out for opportunities to factorize an expression, add fractions, square a binomial or create a monomial denominator.
  3. Look out for opportunities to use the fundamental identities. Always work for the function that you want to arrive at. Sines and cosines pair up well, so do secants and tangents and cosecants and cotangents.
  4. If in case all these do not help, try converting all the functions to sines and cosines.
  • For identities which are of the rational form i.e. form and if the denominator is any one of 1 – cos θ , 1 + cosθ , 1 + sinθ , 1 - sinθ , secθ - 1, secθ + 1, cosecθ - 1, cosecθ + 1, secθ - tanθ , secθ + tanθ , cosecθ - cotθ and cosecθ + cotθ , then we multiply the numerator and denominator by the conjugate of the same.

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