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In our high school classes, we all come across this interesting branch of mathematics, called trigonometry. Trigonometric identities are equalities using trigonometric functions that hold for any value of the variables involved, hence defining both sides of the equality.

We will look at trigonometric identities in this mini-lesson. Sin, cos, and tan are the three main trigonometric ratios. The reciprocals of sin, cos, and tan are represented by the three additional trigonometric ratios sec, cosec, and cot in trigonometry. It might look effortless initially but is not so.

It takes a lot of time and effort to fully master trigonometry. Sadly, our tests and assignments do not wait for us and have deadlines. In that case, you can go for BookMyEssay’s Trigonometric Identities assignment help to make your assignments of trigonometry with ease and within the stipulated time.


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What are Trigonometric Identities?

In mathematics, an "identity" is an equation that is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. Trigonometric identities are equations that relate to various trigonometric functions and are true for every variable value in the domain. Identity is essentially an equation that holds for all possible values of the variable(s) in it. Some algebraic identities are, for example, (a + b).

2 = a2 + 2ab + b2

(a - b)

2 = a2 - 2ab+ b2

(a + b)

(a-b)= a2 - b2

The algebraic identities only apply to the variables, but the trigonometric identities pertain to the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. All of these and the rest of the types are described in detail in our Trigonometric Identities homework help service.

A Few Types of Trigonometric Identities

Reciprocal Trigonometric Identities

Sin, cosine, and tangent are reciprocals of cosecant, secant, and cotangent, respectively.As a result, the reciprocal identities are as follows:

  • sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ
  • cos θ = 1/secθ (OR) sec θ = 1/cosθ
  • tan θ = 1/cotθ (OR) cot θ = 1/tanθ

Pythagorean Trigonometric Identities

The Pythagorean trigonometric identities are derived from Pythagoras' theorem in trigonometry. By applying Pythagoras' theorem to the right-angled triangle below, we get:

  • Opposite+ Adjacent= Hypotenuse2
  • Dividing both sides by Hypotenuse2
  • Opposite2/Hypotenuse2+ Adjacent2/Hypotenuse2 = Hypotenuse2/Hypotenuse2
  • sin2θ + cos2θ = 1

Similarly, we can derive two more Pythagorean trigonometric identities:

  • 1 + tan2θ = sec2θ
  • 1 + cot2θ = cosec2θ


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Complementary and Supplementary Trigonometric Identities

A complementary angle is a pair of two angles whose total is equal to 90°. Angles can be represented by (90 -). The complementary angle trigonometric ratios are as follows:

  • sin (90°- θ) = cos θ
  • cos (90°- θ) = sin θ
  • cosec (90°- θ) = sec θ
  • sec (90°- θ) = cosec θ
  • tan (90°- θ) = cot θ
  • cot (90°- θ) = tan θ

The supplementary angle is composed of two additional angles that add up to 180°. An angle's supplement is (180 -). The following are supplementary angle trigonometric ratios:

  • sin (180°- θ) = sinθ
  • cos (180°- θ) = -cos θ
  • cosec (180°- θ) = cosec θ
  • sec (180°- θ)= -sec θ
  • tan (180°- θ) = -tan θ
  • cot (180°- θ) = -cot θ

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