Trigonometric Identities

Trigonometry is that branch of Mathematics, which relates to the study of angles, measurement of angles, and units of measurement. It also concerns itself with the six ratios for a given angle and the relations satisfied by these ratios. In an extended way, the study is also of the angles forming the elements of a triangle. Logically, a discussion of the properties of a triangle; solving a triangle, physical problems in the area of heights and distances using the properties of a triangle – all constitute a part of the study. It also provides a method of solution of trigonometric equations.

### Trigonometric Identity

An equation involving trigonometric ratios of an angle is called trigonometric Identity if it is true for all values of the angle. These are useful whenever trigonometric functions are involved in an expression or an equation. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side.

Proof of the Trigonometric Identities

For any acute angle θ, prove that
(i) tanθ = sinθ/cosθ
(ii) cotθ = cosθ/sinθ
(iii) tanθ . cotθ = 1
(iv) sin2θ + cos2θ = 1
(v) 1 + tan2θ = sec2θ
(vi) 1 + cot2θ = cosec2θ

Proof:
Consider a right-angled △ABC (fig. 1) in which ∠B = 90° and ∠A = 0°.
Let AB = x units, BC y units and AC = r units.
Then,
(i) tanθ = y/x = (y/r)/(x/r)     [dividing num. and denom. by r]
∴ tanθ = sinθ/cosθ
(ii) cotθ = x/y = (x/r)/(y/r)     [dividing num. and denom. by r]
∴ cotθ = cosθ/sinθ
(iii) tanθ . cotθ  = (sinθ/cosθ) . (cosθ/sinθ)
tanθ . cotθ = 1
Then, by Pythagoras’ theorem, we have
x2 + y= r2.
Now,
(iv) sin2θ + cos2θ  = (y/r)2 + (x/r)= ( y2/r2 + x2/r2)

= (x+ y2)/r2 = r2/r= 1 [x2+ y2 = r2]
sin2θ + cos2θ = 1
(v) 1 + tan2θ = 1 + (y/x)= 1 + y2/x2 = (y+ x2)/x= r2/x2 [x+ y= r2]
(r/x)= sec2θ
∴ 1 + tan2θ = sec2θ.
(vi) 1 + cot2θ = 1 + (x/y)= 1 + x2/y2 = (x+ y2)/y= r2/y2 [x+ y= r2]
(r2/y2) = cosec2θ
∴ 1 + cot2θ = cosec2θ.

Application of Trigonometric Identites

Application 1: Prove that (1 – sin2θ) sec2θ = 1
Proof:
We have:
LHS = (1 – sin2θ) sec2θ
= cos2θ . sec2θ
= cos2θ . (1/cos2θ)
=1
= RHS.
∴ LHS = RHS.

Application 2: Prove that (1 + tan2θ) cos2θ = 1
Proof:
We have:
LHS = (1 + tan2θ)cos2θ
= sec2θ . cos2θ
= (1/cos2θ) . cos2θ
= 1 = RHS.
∴ LHS=RHS.

Application 3: Prove that (cosec2θ – 1) tan²θ  = 1
Proof:
We have:
LHS = (cosec²θ – 1) tan2θ
= (1 + cot2θ – 1) tan2θ
= cot2θ . tan2θ
= (1/tan2θ) . tan2θ
= 1 = RHS.
∴ LHS=RHS.

Application 4: Prove that (sec4θ – sec2θ) = (tan2θ + tan4θ)
Proof:
We have:
LHS = (sec4θ – sec2θ)
= sec2θ(sec2θ – 1)
= (1 + tan2θ) (1 + tan2θ – 1)
= (1 + tan2θ) tan2θ
= (tan2θ + tan4θ)
= RHS
∴ LHS = RHS.

Application 5: Prove that √(sec2θ + cosec2θ) = (tanθ + cotθ)
Proof:
We have:
LHS = √(sec2θ + cosec2θ ) = √((1 + tan2θ) + (1 + cot2θ))
= √(tan2θ + cot2θ + 2)
= √(tan2θ + cot2θ + 2tanθ.cotθ )         (tanθ . cotθ = 1)
= √(tanθ + cotθ)2
= tanθ + cotθ = RHS
∴ LHS = RHS.

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## Mathematics for SSS 1

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## FAQs on Trigonometric Identities - Mathematics for SSS 1

 1. What are trigonometric identities?
Trigonometric identities are mathematical equations that relate the angles and sides of a right triangle. They are used to simplify and manipulate trigonometric expressions and equations.
 2. What is the importance of trigonometric identities?
Trigonometric identities are important because they allow us to solve complex trigonometric problems by transforming them into simpler forms. They also help in proving various mathematical theorems and properties related to triangles and circles.
 3. How do I prove trigonometric identities?
To prove a trigonometric identity, you need to manipulate one side of the equation using known trigonometric identities and properties until it becomes equal to the other side. This can involve simplifying expressions, using trigonometric ratios, applying Pythagorean identities, or using sum and difference formulas.
 4. What are some commonly used trigonometric identities?
Some commonly used trigonometric identities include the Pythagorean identities (sin²θ + cos²θ = 1), sum and difference formulas (sin(A ± B) = sinAcosB ± cosAsinB), double-angle formulas (sin2θ = 2sinθcosθ), and reciprocal identities (cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ).
 5. How can I apply trigonometric identities in real-life situations?
Trigonometric identities are applied in various fields such as physics, engineering, architecture, and navigation. For example, they are used in calculating the height of buildings, designing bridges, analyzing sound waves, and determining the position of objects using GPS systems. By applying trigonometric identities, these real-life problems can be solved accurately and efficiently.

## Mathematics for SSS 1

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