Trigonometric identities are equalities which are involved in trigonometric functions and are true for every single value of occurring mathematical variables. The Trigonometric identity is involved in certain functions of one or more angles in geometry. Trigonometric identity is totally distinct from triangle identities.
Trigonometric identities are used where the expressions involving trigonometric functions need to be simplified. The important application of trigonometric identity is in an integration of non-trigonometric function, by using substitution rule with a trigonometric function, and then the resulting integral is simplified with a trigonometric identity.
The trigonometric functions specify the relationship between the interior angles of right triangle and side length. For example the sine of angle “Ѳ” is defined as the opposite side length divided by the hypotenuse length.
With reference to the above fig, the elementary trigonometry identities are,
Sin Ѳ = opposite / hypotenuse
= a/h
Cos Ѳ = adjacent / hypotenuse
= b/h
Tan Ѳ = opposite / adjacent
= a/b
Cot Ѳ = adjacent / opposite
= b/a
Sec Ѳ = hypotenuse / adjacent
= h/b
Csc Ѳ = hypotenuse / opposite
= h/a
Pythagorean Identities
sin^2Ѳ + cos^2Ѳ = 1
sin^2Ѳ = 1-cos^2Ѳ
cos^2Ѳ = 1-sin^2Ѳ
1 + tan^2Ѳ = sec^2Ѳ
1 + cot^2Ѳ = csc^2 Ѳ
Quotient Identities
TanѲ = sinѲ / cosѲ
Cot Ѳ = cos Ѳ / sin Ѳ
Trigonometric Identities Proof
Trigonometric identity proof is used to show the relationship between trigonometric functions. Trigonometric identity proofs are different from solving the trigonometric identity. The identities mean that the “tautology” or “equality”, the statement or an equation that is always true for any value of the variable. But an equation is an equality that is true only for certain specific values of the variables.
Example:
In algebraic, (x + 5) (x + 5) = (x^2 – 25)
Proof of Reciprocal identities
Sin Ѳ = 1/Csc Ѳ
Sin Ѳ = opposite / hypotenuse
= a/h
Csc Ѳ = hypotenuse / opposite
= h/a
a/h = 1/h/a
Hence proved
According to the same proof like above we get the following reciprocal identities
Cos Ѳ = 1/sec Ѳ
Tan Ѳ = 1/cot Ѳ
Csc Ѳ = 1/sin Ѳ
Sec Ѳ = 1/ cos Ѳ
Cot Ѳ = 1/tan Ѳ
Identities are that, in calculation, we can replace any member of the identity with other any member of identity. So in above example Sin Ѳ can be replaced with 1 /Csc Ѳ or 1 /Csc Ѳ can be replaced with Sin Ѳ.
Trigonometric Identities Examples
Sec 2 x + csc2 x = sec^2 x. csc2 x
We know that sec^2 x= 1/cos^2x
csc2 x = 1/sin^2x
Sec 2 x + csc2 x = 1/cos^2x + 1/sin^2x
= sin^2 x + cos^2x/ cos^2x sin^2x (sin^2x + cos^2x =1)
= 1/cos^2x.sin^2x (1/cos^2x = sec^2x, 1/sin^2x = csc2x)
= sec^2 x. csc2 x
Solving Trigonometric Identities
cot Ѳ / csc Ѳ = cos Ѳ
We know that,
Cot Ѳ = cos Ѳ/ sin Ѳ
Csc Ѳ = 1/sinѲ
cotѲ/cscѲ = (cos Ѳ/sinѲ)/ (1/sinѲ)
= (cos v/sinѲ) X (sinѲ/1)
= cosѲ
cotѲ + tan Ѳ = sec Ѳ . cscѲ
We know that,
cot Ѳ = cos Ѳ/sinѲ
tan Ѳ = sinѲ/cosѲ
So,
cotѲ + tan Ѳ = (cos Ѳ/sinѲ) + (sinѲ/cosѲ)
= (cos^2 Ѳ/sin Ѳ.cosѲ) + (sin^2 Ѳ/sin Ѳ.cosѲ)
= sin 2Ѳ + cos^2Ѳ / sin Ѳ. cos Ѳ
= (1/sinѲ).(1/cos Ѳ) {1/sinѲ=secѲ, cscѲ=1/cosѲ}
= sec Ѳ .cscѲ
Trigonometric identities are used where the expressions involving trigonometric functions need to be simplified. The important application of trigonometric identity is in an integration of non-trigonometric function, by using substitution rule with a trigonometric function, and then the resulting integral is simplified with a trigonometric identity.
The trigonometric functions specify the relationship between the interior angles of right triangle and side length. For example the sine of angle “Ѳ” is defined as the opposite side length divided by the hypotenuse length.
With reference to the above fig, the elementary trigonometry identities are,
Sin Ѳ = opposite / hypotenuse
= a/h
Cos Ѳ = adjacent / hypotenuse
= b/h
Tan Ѳ = opposite / adjacent
= a/b
Cot Ѳ = adjacent / opposite
= b/a
Sec Ѳ = hypotenuse / adjacent
= h/b
Csc Ѳ = hypotenuse / opposite
= h/a
Pythagorean Identities
sin^2Ѳ + cos^2Ѳ = 1
sin^2Ѳ = 1-cos^2Ѳ
cos^2Ѳ = 1-sin^2Ѳ
1 + tan^2Ѳ = sec^2Ѳ
1 + cot^2Ѳ = csc^2 Ѳ
Quotient Identities
TanѲ = sinѲ / cosѲ
Cot Ѳ = cos Ѳ / sin Ѳ
Trigonometric Identities Proof
Trigonometric identity proof is used to show the relationship between trigonometric functions. Trigonometric identity proofs are different from solving the trigonometric identity. The identities mean that the “tautology” or “equality”, the statement or an equation that is always true for any value of the variable. But an equation is an equality that is true only for certain specific values of the variables.
Example:
In algebraic, (x + 5) (x + 5) = (x^2 – 25)
Proof of Reciprocal identities
Sin Ѳ = 1/Csc Ѳ
Sin Ѳ = opposite / hypotenuse
= a/h
Csc Ѳ = hypotenuse / opposite
= h/a
a/h = 1/h/a
Hence proved
According to the same proof like above we get the following reciprocal identities
Cos Ѳ = 1/sec Ѳ
Tan Ѳ = 1/cot Ѳ
Csc Ѳ = 1/sin Ѳ
Sec Ѳ = 1/ cos Ѳ
Cot Ѳ = 1/tan Ѳ
Identities are that, in calculation, we can replace any member of the identity with other any member of identity. So in above example Sin Ѳ can be replaced with 1 /Csc Ѳ or 1 /Csc Ѳ can be replaced with Sin Ѳ.
Trigonometric Identities Examples
Sec 2 x + csc2 x = sec^2 x. csc2 x
We know that sec^2 x= 1/cos^2x
csc2 x = 1/sin^2x
Sec 2 x + csc2 x = 1/cos^2x + 1/sin^2x
= sin^2 x + cos^2x/ cos^2x sin^2x (sin^2x + cos^2x =1)
= 1/cos^2x.sin^2x (1/cos^2x = sec^2x, 1/sin^2x = csc2x)
= sec^2 x. csc2 x
Solving Trigonometric Identities
cot Ѳ / csc Ѳ = cos Ѳ
We know that,
Cot Ѳ = cos Ѳ/ sin Ѳ
Csc Ѳ = 1/sinѲ
cotѲ/cscѲ = (cos Ѳ/sinѲ)/ (1/sinѲ)
= (cos v/sinѲ) X (sinѲ/1)
= cosѲ
cotѲ + tan Ѳ = sec Ѳ . cscѲ
We know that,
cot Ѳ = cos Ѳ/sinѲ
tan Ѳ = sinѲ/cosѲ
So,
cotѲ + tan Ѳ = (cos Ѳ/sinѲ) + (sinѲ/cosѲ)
= (cos^2 Ѳ/sin Ѳ.cosѲ) + (sin^2 Ѳ/sin Ѳ.cosѲ)
= sin 2Ѳ + cos^2Ѳ / sin Ѳ. cos Ѳ
= (1/sinѲ).(1/cos Ѳ) {1/sinѲ=secѲ, cscѲ=1/cosѲ}
= sec Ѳ .cscѲ
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