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Trigonometric Ratio
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Trigonometric Ratio: Trigonometric ratios of the angles are the ratios of the sides of a triangle with respect to its acute angles.

Trigonometric Identities: An equation involving trigonometric ratios of an angle (say) is said to be a trigonometric identity if it is satisfied for all values of for which the given trigonometric ratios are defined.

Here we will discuss how to use trigonometric identities in proving other trigonometric identities with help of some examples.

Example: Prove the following identities:

(i)cos4A - cos2A = sin4A - sin2A
(ii)sin4A + cos4A = 1 - 2 sin2A cos2 A

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(i) Given:
= cos4A - cos2A
= cos2A (cos2A - 1)
= -cos2A (1 - cos2A)
= -cos2A sin2A
= -1(1 - sin2A) sin2A
= -sin2A + sin4A
= sin4A - sin2A = RHS

(ii) Given:
= sin4A+ cos4A
= (sin2A)2 (cos2A)2+ 2 sin2A cos2A - 2 sin2Acos2A
= (sin2A + cos2A)2 - 2 sin2A cos2A
= 1 - 2 sin2A cos2A = RHS
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