# Trigonometric Identities for Class 10th – Formulas

Here, you will learn various trigonometric identities for class 10th and formulas of trigonometry.

Let’s begin-

In a right angle triangle

sin$$\theta$$ = $$p\over h$$; cos$$\theta$$ = $$b\over h$$; tan$$\theta$$ = $$p\over b$$; cosec$$\theta$$ = $$h\over p$$; sec$$\theta$$ = $$h\over b$$ and cot$$\theta$$ = $$b\over p$$

where ‘p’ is perpendicular ; ‘b’ is base and ‘h’ is hypotenuse.

## Basic Trigonometric Identities for Class 10th :

(1)  $$sin\theta$$.$$cosec\theta$$ = 1

(2)  $$cos\theta$$.$$sec\theta$$ = 1

(3)  $$tan\theta$$.$$cot\theta$$ = 1

(4)  $$tan\theta$$ = $$sin\theta\over{cos\theta}$$   $$cot\theta$$ = $$cos\theta\over{sin\theta}$$

(5)  $$sin^2\theta$$ + $$cos^2\theta$$ = 1

(6)  $$sec^2\theta$$ – $$tan^2\theta$$ = 1

(7)  $$cosec^2\theta$$ – $$cot^2\theta$$ = 1

Trigonometric Ratios of the sum & difference of two angles :

(1)   sin(A + B) = sin A cos B + cos A sin B

(2)   sin(A – B) = sin A cos B – cos A sin B

(3)   cos(A + B) = cos A cos B – sin A sin B

(4)   cos(A – B) = cos A cos B + sin A sin B

(5)   tan(A + B) = $$tan A + tan B\over {1 – tan A tan B}$$

(6)   tan(A – B) = $$tan A – tan B\over {1 + tan A tan B}$$

(7)   cot(A + B) = $$cot B cot A – 1\over {cot B + cot A}$$

(8)   cot(A – B) = $$cot B cot A + 1\over {cot B – cot A}$$

Formulae to transform the product into sum or difference :

(i)   2 sin A cos B = sin(A + B) + sin(A – B)

(ii)   2 cos A sin B = sin(A + B) – sin(A – B)

(iii)   2 cos A cos B = cos(A + B) – cos(A – B)

(iv)   2 sin A sin B = cos(A – B) – cos(A + B)

Formulae to transform the sum or difference into product :

(i)    sin C + sin D = 2 sin($$C + D\over 2$$) cos($$C – D\over 2$$)

(ii)   sin C – sin D = 2 cos($$C + D\over 2$$) sin($$C – D\over 2$$)

(iii)  cos C + cos D = 2 cos($$C + D\over 2$$) cos($$C – D\over 2$$)

(iv)  cos C – cos D = 2 sin($$C + D\over 2$$) sin($$D – C\over 2$$)

Trigonometric ratios of sum of more than two angles :

(i)   sin(A + B + C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB – sinAsinBsinC

(ii)  cos(A + B + C) = cosAcosBcosC – sinAsinBcosC – sinAcosBsinC – cosAsinBsinC

(iii)  tan(A + B + C) = $$tanA + tanB + tanC – tanAtanBtanC\over {1 – tanAtanB – tanBtanC – tanCtanA}$$

Trigonometric ratios of mutiple angles :

(i)  sin2A = 2sinAcosA = $$2tanA\over {1+tan^2A}$$

(ii)  cos2A = $$cos^2A$$ – $$sin^2A$$ = $$2cos^2A$$ – 1 = 1 – $$2sin^2A$$ = $$1 – tan^2A\over {1 + tan^A}$$

(iii)  1 + cos2A = $$2cos^2A$$

(iv)  1 – cos2A = $$2sin^2A$$

(v)   tanA = $$1 – cosA\over {sin2A}$$ = $$sin2A\over {1+cos2A}$$

(vi)  tan2A = $$2tanA\over {1-tan^2A}$$

(vii)  sin3A = 3sinA – $$4sin^3A$$

(viii)  cos3A = $$4cos^3A$$ – 3cosA

(ix)  tan3A = $$3tanA – tan^3A\over {1 – 3tan^2A}$$