## Trigonometrical Identities:

An equation involving trigonometric ratios of an angle θ (say), which is true for all values of θ for which the given trigonometric ratios are defined, is called trigonometric identities.

The situation is exactly similar to algebraic identity viz; (a +b)^{2} = a^{2}+ 2ab + b^{2} which is true for all values of a and b.

**Example I-** cos^{2} θ – 1/2 cos θ = cos θ (cos θ – 1/2) is a trigonometric identity, whereas cos θ (cos θ – 1/2) = 0 is an equation.

**Example II-** cosec θ = 1/sinθ is a trigonometric identity because it holds for all values of θ except for which sin θ = 0.

For sin θ = 0, cosec θ is not defined.

Now we shall establish the following two theorems which form the backbone of any trigonometrical identity.

Theorem I: For any acute angle θ, prove that the following identities- (i) sin^{2} θ + cos^{2} θ = 1 (ii) 1 + tan^{2} θ = sec^{2} θ (iii) 1 + cot^{2} θ = cosec^{2} θ.Proof: Suppose an acute angle ∠XAY = θ has been given. Take a point P on AY. Draw PM ⊥ AX.In right-angled triangle AMP, let Base = AM = x, Perepndicular = PM = y and Hypotenuse = AP = r. Then, by the Pythagoras theorem, we have x ^{2} + y^{2} = r^{2}.Now (i) sin ^{2} θ + cos^{2} θ = (y/r)^{2} + (x/r)^{2} = y^{2}/r^{2} + x^{2}/r^{2} = y^{2}+x^{2}/r^{2} = r^{2}/r^{2} = 1 [∵ x^{2} + y^{2} = r^{2}]Hence sin ^{2} θ + cos^{2} θ = 1.(ii) 1 + tan ^{2} θ = 1 + (y/x)^{2} = 1 + y^{2}/x^{2 } = x^{2}+y^{2}/x^{2} = r^{2}/x^{2} = (r/x)^{2 }= sec^{2} θ [∵ x^{2} + y^{2} = r^{2}]Hence 1 + tan ^{2} θ = sec^{2} θ.(iii) 1 + cot ^{2} θ = 1 + (x/y)^{2} = 1 + x^{2}/y^{2} = y^{2}+x^{2}/y^{2} = r^{2}/y^{2} = (r/y)^{2} = cosec^{2} θ [∵ x^{2} + y^{2} = r^{2}]Hence 1 + cot ^{2} θ = cosec^{2} θ. |

Theorem II: For any acute angle θ, prove the identities: (i) tan θ = sin θ/cos θ (ii) cot θ = cos θ/sin θ (iii) tan θ cot θ = 1.Proof: As given in the above figure, we have(i) tan θ = y/x = (y/r)/(x/r) = sin θ/cos θ [Dividing numerator and denominator by r] ∴ tan θ = sin θ/cos θ. (ii) cot θ = x/y = (x/r)/(y/r) = cos θ/sin θ [Dividing numerator and denominator by r] ∴ cot θ = cos θ/sin θ. (iii) tan θ cot θ = y/x . x/y = 1 ∴ tan θ cot θ = 1. |

**Important:** As a consequence of the identities given in theorem I, we have-

(i) 1 – sin^{2} θ = cos^{2} θ and 1 – cos^{2} θ = sin^{2} θ.

(ii) sec^{2} θ – tan^{2} θ = 1 and sec^{2} θ – 1 = tan^{2} θ.

(iii) cosec^{2} θ – cot^{2} θ = 1 and cosec^{2} θ – 1 = cot^{2} θ.

Since the proofs of identities of theorem I involve Pythagoras Theorem, they are called **Pythagorean identities.**

The identity sin^{2} θ + cos^{2} θ = 1 tells us that the point P with coordinates (cos θ, sin θ) lies on the unit circle x^{2} + y^{2} = 1, i.e., the circle with origin as the centre and unity as the radius.

Since by definition, cos θ and sin θ are both non-negative i.e., ≥ 0 for 0° ≤ θ ≤ 90°, P is in the first quadrant of the circle. Conversely, if we take a point P in the first quadrant of the unit circle, then the coordinates of P are (OM, MP) i.e., (OM/OP, MP/OP), since OP = 1 i.e., (cos θ, sin θ) where θ is the angle XOP.

It is for this reason that the trigonometric ratios cos θ and sin θ are also called circular functions.