Quotient Identities

Trigonometry is a branch of mathematics in which we study the relationship between the angles and sides of triangles. In trigonometry, there are various identities that relate the trigonometric functions to each other, such as the Pythagorean identities, quotient identities, reciprocal identities, etc.

Quotient Identity Formula

Quotient identities are fundamental trigonometric identities that relate the tangent and cotangent functions to the sine and cosine functions. Specifically, they define the ratio (quotient) of sine to cosine for tangent, and cosine to sine for cotangent.

• Tangent Identity: tan⁡ θ = sin ⁡θ/cos ⁡θ.
• Cotangent Identity: cot⁡ θ = cos⁡ θ/sin⁡ θ.

Proof of Quotient Identities

Identity: tan ⁡θ = sin ⁡θ/cos ⁡θ.

As we know, \tan \theta = \dfrac{\text{Perpendicular}}{\text{Base}}

Dividing numerator and denominator by hypotenuse. we get,

\tan\theta = \dfrac{\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}}{\dfrac{\text{Base}}{\text{Hypotenuse}}}

By definition,\sin \theta = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}} \text{ and } \cos \theta = \dfrac{\text{Base}}{\text{Hypotenuse}}

\tan \theta = \frac{\sin \theta}{\cos \theta}

Hence, we can say that tan⁡ θ = sin ⁡θ/cos ⁡θ.

Similarly, cot⁡ θ = cos ⁡θ/sin ⁡θ.

As we know, \cot \theta = \dfrac{\text{Base}}{\text{Perpendicular}}

Dividing the numerator and denominator by the hypotenuse, we get:

\cot \theta = \dfrac{\dfrac{\text{Base}}{\text{Hypotenuse}}}{\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}}

By definition, \cos \theta = \dfrac{\text{Base}}{\text{Hypotenuse}} \quad \text{and} \quad \sin \theta = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}}

Thus, \cot \theta = \dfrac{\cos \theta}{\sin \theta}

Hence, we can say that \cot \theta = \dfrac{\cos \theta}{\sin \theta}.

Solved Example On Quotient Identities

Example 1: Simplify the expression sin ⁡θ/cos ⁡θ + cos ⁡θ/sin ⁡θ.

Solution:

Given: sin ⁡θ/cos ⁡θ + cos ⁡θ/sin ⁡θ
= \dfrac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}

Using the Pythagorean identity \sin^2 \theta + \cos^2 \theta = 1, we get
= \dfrac{1}{\sin \theta \cos \theta} = \dfrac{1}{\sin \theta} \times \dfrac{1}{\cos \theta} = \csc \theta \times \sec \theta

Thus, sin ⁡θ/cos ⁡θ + cos ⁡θ/sin ⁡θ = cosec θ sec θ

Example 2: Prove the identity tan⁡²θ + 1 = sec⁡²θ.

Solution:

Given: tan⁡²θ + 1 = sec⁡²θ

Using the quotient identity tan ⁡θ = sin ⁡θ/cos ⁡θ​, we know that:
tan⁡²θ = sin⁡²θ/cos^⁡2θ

LHS = sin⁡²θ/cos^⁡2θ + 1
LHS = (sin⁡²θ + cos^⁡2θ)/cos^⁡2θ

By the Pythagorean identity sin^⁡2θ + cos⁡²θ = 1, dividing both sides by cos^⁡2θ gives:
LHS = 1/cos^⁡2θ = sec⁡²θ

Thus, the identity is proven.

Example 3: Prove that cot^⁡2θ + 1 = csc⁡²θ.

Solution:

Given: cot^⁡2θ + 1 = csc⁡²θ

Using the quotient identity cot ⁡θ = cos ⁡θ/sin ⁡θ​, we know that:
cot⁡²θ = cos⁡²/θsin⁡²θ.

LHS = cot^⁡2θ + 1 = cos⁡²/θsin⁡²θ + 1 = (cos⁡² θ+ sin⁡²θ)/sin⁡²θ

By the Pythagorean identity sin^⁡2θ + cos⁡²θ = 1, dividing both sides by sin^⁡2θ gives:
LHS = 1/sin⁡²θ = csc⁡²θ.

Thus, the identity is proven.

Example 4: Simplify tan ⁡θ/cot ⁡θ.

Solution:

Using the quotient identities tan ⁡θ = sin ⁡θ/cos ⁡θ and cot ⁡θ = cos ⁡θ/sin ⁡θ​:
tan ⁡θ/cot ⁡θ = sin ⁡θ/cos ⁡θ/cos ⁡θ/sin ⁡θ = sin⁡²θ/cos⁡²θ

Thus, the simplified expression is tan⁡²θ.

Also Read: Trigonometry In Math

Practice Questions on Quotient Identities

You can download a free worksheet on Quotient Identities for practicing various different questions with their answers from below:

Conclusion

Quotient identities, particularly the identities for tangent and cotangent, are powerful tools for simplifying trigonometric expressions and solving equations. They form part of the foundation of trigonometric simplifications, allowing for easier manipulation of complex terms and helping in various applications like calculus, geometry, and engineering.

Read More,

• Trigonometry in Maths
• Trigonometry Table
• Trigonometric Identities