Trigonometric Identities

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A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angle θ for which the ratios are defined. Unlike equations that are true for specific values,

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sin²θ + cos²θ = 1 This identity states that the sum of squares of sine and cosine of any angle is always equal to 1. It is derived from the Pythagorean theorem in a right triangle.

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1 + tan²θ = sec²θ This identity relates tangent and secant functions. It can be derived by dividing the first identity (sin²θ + cos²θ = 1) by cos²θ throughout.

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1 + cot²θ = cosec²θ This identity relates cotangent and cosecant functions. It can be derived by dividing the first identity (sin²θ + cos²θ = 1) by sin²θ throughout.

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In a right triangle ABC with right angle at B: sin θ = AB/AC (perpendicular/hypotenuse) cos θ = BC/AC (base/hypotenuse) sin²θ + cos²θ = (AB/AC)² + (BC/AC)² = (AB² + BC²)/AC² = AC²/AC² [By Pythagoras

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Starting with sin²θ + cos²θ = 1 Divide both sides by cos²θ: (sin²θ)/cos²θ + (cos²θ)/cos²θ = 1/cos²θ tan²θ + 1 = sec²θ Therefore: 1 + tan²θ = sec²θ This uses the fact that tan θ = sin θ/cos θ and s

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Starting with sin²θ + cos²θ = 1 Divide both sides by sin²θ: (sin²θ)/sin²θ + (cos²θ)/sin²θ = 1/sin²θ 1 + cot²θ = cosec²θ This uses the fact that cot θ = cos θ/sin θ and cosec θ = 1/sin θ.

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From the identity sin²θ + cos²θ = 1: sin²θ = 1 - cos²θ sin θ = ±√(1 - cos²θ) For angles between 0° and 90° (which is our focus in Class 10), sin θ is positive, so: sin θ = √(1 - cos²θ)