# Trigonometric Identities - Class 10

Trigonometry is a branch of Mathematics that deals with the study of measuring the sides and angles of a triangle. The concept of trigonometry is completely based on right triangles. There are six trigonometric ratios namely sine, cosine, tangent, cotangent, secant, and cosecant of a reference angle. All these trigonometric ratios are expressed as the ratios of the hypotenuse, base and perpendicular side of a right triangle. Trigonometric identities class 10 establishes the relationship between different trigonometric ratios. These trigonometric identities are the basic formulae that are true for all the values of the reference angle.

Trigonometric Identities Class 10 List:

Class 10 describes a few trigonometric identities which can be proved with the basic knowledge of trigonometry. It is the most interesting part of Class 10 trigonometry. Trigonometric identities can be used to solve trigonometric problems easily by reducing the number of computational steps. All the trigonometric identities class 10 questions can be easily answered with the knowledge of these identities.

 Sin2A + Cos2A = 1 Sec2A = 1 + Tan2A Cosec2A = 1 + Cot2A

In the above equations, ‘A’ is the reference angle for which the trigonometric ratios are written.

Basic Trigonometric Ratios:

There are six trigonometric ratios namely sine, cosine, tangent cotangent, secant, and cosecant. The definitions of each trigonometric ratio for a reference angle ‘θ’ is mentioned in terms of the sides of a right triangle.

• The sine of an angle θ is the ratio of the opposite side to the hypotenuse.

• Cosine of an angle θ is the ratio of the adjacent side to the hypotenuse.

• Tangent of an angle θ is the ratio of the opposite side to the adjacent side.

• The cotangent of the angle θ is the ratio of the adjacent side to the opposite side.

• The secant of an angle θ is the ratio of the hypotenuse to the adjacent side.

• Cosecant of the angle θ is the ratio of the hypotenuse to the opposite side.

Derivation of Trigonometric Identities Class 10:

Let us consider a right triangle ABC whose sides are AB, BC, and AC respectively. The triangle is right-angled at C. Considering ‘α’ as the reference angle, the opposite side is BC and the adjacent side is AC. AB is the longest side of the right triangle and is called the hypotenuse.

Applying Pythagorean theorem to the above triangle,

AB² = AC²  + BC²

On dividing the entire equation by the hypotenuse i.e. AB, we get,

$\frac{AB²}{AB²}$ = $\frac{AC²}{AB²}$ + $\frac{BC²}{AB²}$

1 = ($\frac{AC}{AB}$)² +  ($\frac{BC}{AB}$)²

By the definition of basic trigonometric functions,

Sin α = $\frac{Opposite}{Hypotenuse}$ = $\frac{BC}{AB}$ and Cos α = $\frac{Adjacent}{Hypotenuse}$ = $\frac{AC}{AB}$

Substituting these values in the above equation, the equation implies,

1 = (cos α)² + (sin α)²

The equation can be rewritten to give the first one among the trigonometric identities class 10 as:

(sin α)² + (cos α)² = 1

Derivation of Trigonometric Identities Class 10 (2):

Let us consider a right triangle ABC whose sides are AB, BC, and AC respectively. The triangle is right-angled at C. Considering ‘α’ as the reference angle, the opposite side is BC and the adjacent side is AC. AB is the longest side of the right triangle and is called the hypotenuse.

Applying Pythagorean theorem to the above triangle,

AB² = AC² + BC²

On dividing the entire equation by the adjacent side i.e. AC, we get,

$\frac{AB²}{AC²}$ = $\frac{AC²}{AC²}$ = $\frac{BC²}{AC²}$

($\frac{AB}{AC}$)² = 1 + ($\frac{BC}{AC}$)²

By the definition of basic trigonometric functions,

tan α = $\frac{Opposite}{Adjacent}$ = $\frac{BC}{AC}$ and Sec α = $\frac{Hypotenuse}{Adjacent}$ = $\frac{AB}{AC}$

Substituting these values in the above equation, the equation implies,

(sec α)² = 1 + (tan α)²

The equation can be rewritten to give the second one among the trigonometric identities class 10 as:

sec²α = 1 + tan²α

Derivation of Trigonometric Identities Class 10 (3):

Let us consider a right triangle ABC whose sides are AB, BC, and AC respectively. The triangle is right-angled at C. Considering ‘α’ as the reference angle, the opposite side is BC and the adjacent side is AC. AB is the longest side of the right triangle and is called the hypotenuse.

Applying Pythagorean theorem to the above triangle,

AB² = AC² + BC²

On dividing the entire equation by the opposite side i.e. BC, we get,

$\frac{AB²}{BC²}$ = $\frac{AC²}{BC²}$ = $\frac{BC²}{BC²}$

($\frac{AB}{BC}$)² = ($\frac{AC}{BC}$)² + 1

By the definition of basic trigonometric functions,

cot α = $\frac{Adjacent}{Opposite}$ =  $\frac{AC}{BC}$ and Cosec α = $\frac{Hypotenuse}{Opposite}$ = $\frac{AB}{BC}$

Substituting these values in the above equation, the equation implies,

(cosec α)² = (cot α)² + 1

The equation can be rewritten to give the third one among the trigonometric identities class 10 as:

cosec² α = 1 + cot² α

Trigonometric Identities Class 10 Problems:

1. Find the value of 1 - Sin2 A

Solution:

1 - Sin2 A = Sin2 A + Cos2 A - Sin2 A = Cos2 A

2. Prove that Sec2P - tan2P - Cosec2P + Cot2P = 0

Solution:

Sec2P - tan2P - Cosec2P + Cot2P = 1 + tan2P - tan2P - (1 + Cot2P) + Cot2P

= 1 + 0 - 1 - Cot2P + Cot2P

= 0

Fun Facts:

• Trigonometric identities are true for all the angles between 00 and 900.

• With the help of these basic trigonometric identities, many other identities are derived.

1. What is the Range of Angles for which Trigonometric Identities Class 10 are True?

Trigonometric identities are true for all angles between 00 and 900. This can be verified as follows.

1. Sin2A + Cos2A = 1

If A = 0, Sin 0 = 0 and Cos 0 = 1

So, Sin2A + Cos2A = 0 + 1 = 1

If A = 900, Sin 900 = 1 and Cos 900 = 0

So, Sin2A + Cos2A = 1 + 0 = 1

1. Sec2A = 1 + Tan2A

If A = 0, Sec 0 = 1 and tan 0 = 0

So, Sec2A = 1 + Tan2A => 1 = 1 + 0 => 1 = 1

If A = 900, Sec 900 = ∞ and tan 900 = ∞

So, Sec2A = 1 + Tan2A =>  ∞ = 1 + ∞ => ∞ = ∞

1. Cosec2A = 1 + Cot2A

If A = 0, Cosec 0 = ∞  and Cot 0 = ∞

So, Cosec2A = 1 + Cot2A => ∞ = 1 + ∞ => ∞ = ∞

If A = 900, Cosec 900 = 1 and Cot 900 = 0

So, Cosec2A = 1 + Cot2A =>  1 = 1 + 0 => 1 = 1

2. What are the Basic Trigonometric Identities Class 10? How are they Derived?

There are three basic trigonometric identities in class 10 which relate the trigonometric ratios mutually. The three basic trigonometric identities learned in class 10 are:

Sin2A + Cos2A = 1

Sec2A = 1 + Tan2A

Cosec2A = 1 + Cot2A

These identities can be derived by considering the right triangle. The right triangle is subjected to the Pythagorean theorem. The equation obtained by applying Pythagoras theorem to the right triangle is divided by the hypotenuse, base, and perpendicular separately. The basic definitions of the trigonometric ratios and the equations obtained on division are compared to get the results of trigonometric identities.