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Reciprocal Identities
Reciprocal Identities are the reciprocals of the six main trigonometric functions, namely sine, cosine, tangent, cotangent, secant, cosecant. The important thing to note is that reciprocal identities are not the same as the inverse trigonometric functions. Every fundamental trigonometric function is a reciprocal of another trigonometric function. For example, cosecant is the reciprocal identity of the sine function.
In this article, we will determine the reciprocal identities, prove the reciprocal identities and find the relationship between them with the help of solved examples.
1.  What are Reciprocal Identities? 
2.  Reciprocal Identities Formulas 
3.  Proof of Reciprocal Identities 
4.  Relationship of Reciprocal Identities 
5.  FAQs on Reciprocal Identities 
What are Reciprocal Identities?
The reciprocals of the six fundamental trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) are called reciprocal identities. The reciprocal identities are important trigonometric identities that are used to solve various problems in trigonometry. Each trigonometric function is a reciprocal of another trigonometric function. The sine function is the reciprocal of the cosecant function and viceversa; the cosine function is the reciprocal of the secant function and viceversa; cotangent function is the reciprocal of the tangent function and viceversa.
Reciprocal Identities Formulas
Reciprocal identities are applied in various trigonometry problems to simplify the calculations. The formulas of the six main reciprocal identities are:
 sin x = 1/cosec x
 cos x = 1/sec x
 tan x = 1/cot x
 cot x = 1/tan x
 sec x = 1/cos x
 cosec x = 1/sin x
Proof of Reciprocal Identities
Now, that we know the reciprocal identities of trigonometry, let us now prove each one of them using the definition of the basic trigonometric functions. First, we will derive the reciprocal identity of the sine function. Consider a rightangled triangle ABC with a right angle at C.
We know that sin θ = Perpendicular/Hypotenuse = c/a and cosec θ = Hypotenuse/Perpendicular = a/c ⇒ sin θ is the reciprocal of cosec θ and cosec θ is the reciprocal of sin θ. Similarly, we will prove other reciprocal identities. cos θ = Base/Hypotenuse = b/a and cosec θ = Hypotenuse/Base = a/b ⇒ cos θ is the reciprocal of sec θ and sec θ is the reciprocal of cos θ. tan θ = sin θ/cos θ and cot θ = cos θ/sin θ ⇒ tan θ is the reciprocal of cot θ and cot θ is the reciprocal of tan θ. Hence, we have
 sin θ is the reciprocal of cosec θ
 cosec θ is the reciprocal of sin θ
 cos θ is the reciprocal of sec θ
 sec θ is the reciprocal of cos θ
 tan θ is the reciprocal of cot θ
 cot θ is the reciprocal of tan θ
Relationship of Reciprocal Identities
As we know that the product of a number and its reciprocal is always equal to one, we have established similar relationships between the reciprocal identities. The product of a trigonometric function and its reciprocal is equal to 1. Hence, we have
 sin θ × cosec θ = 1
 cos θ × sec θ = 1
 tan θ × cot θ = 1
The above equations establish a relationship between the reciprocal identities of trigonometry for any angle θ and show that the product of a trigonometric function and its reciprocal is equal to 1.
Important Notes on Reciprocal Identities
 sin θ is the reciprocal of cosec θ
 cosec θ is the reciprocal of sin θ
 cos θ is the reciprocal of sec θ
 sec θ is the reciprocal of cos θ
 tan θ is the reciprocal of cot θ
 cot θ is the reciprocal of tan θ
Related Topics on Reciprocal Identities
Reciprocal Identities Examples

Example 1: Simplify the expression [cot x(sin x + tan x)/(cosec x + cot x)] using reciprocal identities.
Solution: We will write the expression [cot x(sin x + tan x)/(cosec x + cot x)] in terms sin x and cos x and use the reciprocal identity cosec x = 1/sin x
[cot x(sin x + tan x)/(cosec x + cot x)] = [(cos x/sin x)(sin x + (sin x/cos x))/((1/sin x) + (cos x/sin x))]
= (cos x + 1)(sin x)/(1 + cos x)
= sin x
Answer: [cot x(sin x + tan x)/(cosec x + cot x)] = sin x

Example 2: Determine the value of sec x if cos x = 3/7 using reciprocal identity.
Solution: We know the reciprocal identity sec x = 1/cos x
So, if cos x = 3/7, then sec x = 1/cos x = 1/(3/7) = 7/3
Answer: sec x = 7/3
FAQs on Reciprocal Identities
What are Reciprocal Identities in trigonometry?
The reciprocals of the six fundamental trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) are called reciprocal identities. The sine function is the reciprocal of the cosecant function and viceversa; the cosine function is the reciprocal of the secant function and viceversa; cotangent function is the reciprocal of the tangent function and viceversa.
What are the Six Reciprocal Identities?
The formulas of the six main reciprocal identities are:
 sin x = 1/cosec x
 cos x = 1/sec x
 tan x = 1/cot x
 cot x = 1/tan x
 sec x = 1/cos x
 cosec x = 1/sin x
What is the Reciprocal Identity of Cos x?
The reciprocal identity of cos x is sec x because cos x = 1/sec x. The secant function is the reciprocal of the cosine function.
When to Use Reciprocal Identities?
The reciprocal identities are very useful when solving trigonometric equations. If you find a way to multiply each side of an equation by a trigonometric function's reciprocal, you may be able to reduce some part of the equation and simplify it.
What is the Reciprocal Identity of Sin x?
The reciprocal identity of sin x is cosec x because sin x = 1/cosec x. The cosecant function is the reciprocal of the sine function.
What is the Difference Between Quotient and Reciprocal Identities?
In trigonometry, quotient identities refer to trigonometric identities that are divided by each other whereas reciprocal identities are ones that are the multiplicative inverses of the trigonometric functions.
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