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Trigonometric Identities
AN IDENTITY IS AN EQUALITY that is true for any value of the variable. (An equation is an equality that is true only for certain values of the variable.)
The significance of an identity is that, in calculation, we may replace either member of the identity with the other. We use an identity to give an expression a more convenient form. In calculus and all its applications, the trigonometric identities are of central importance.
In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables .Geometrically; these are identities involving certain functions of one or more angles. These are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. The point about an identity is that, in calculation, we may replace either member of the identity with the other. Thus if we see
"sinθ",then we may,if we wish,replace it with "1/cscθ "and symmetrically,if we see "1/cscθ ",then we may replace it with "sinθ ".
The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:cos2θ+sin2θ = 1 ,Where sin2x means (sin(x))2. This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 = 1 for the unit circle. This equation can be solved for either the sine or the cosine:
Related identities
Dividing the Pythagorean identity through by either cos2 θ or sin2 θ yields two other identities:
1 + tan2θ = sec2θ and 1 + cot2θ = csc2θ
Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other .
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