Question

$\fcolorbox{blue}{red}{derivation \: of \: } \\ = \green{\boxed{ \red{ {sin}^{2} a + {cos}^{2} a = 1}}} \\ \\ \sf{quality \: answers \: needed} \\ \\ \sf{unsatisfying \: answer \: will \: be \: reported}$
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Answer 1

Question :

To derive : sin²θ + cos²θ = 1

Derivation :

• Pythagoras theorem : Let p and b be the legs of the right angled triangle and h be it's hypotenuse , then p² + b² = h² .
• sinθ = p/h
• cosθ = b/h
• tanθ = p/b
• cosecθ = h/p
• secθ = h/b
• cotθ = b/p

According to the Pythagoras theorem ,

We have ,

p² + b² = h² ----------(1)

Now ,

Dividing eq-(1) by h² , we get ;

=> (p² + b²)/h² = h²/h²

=> p²/h² + b²/h² = 1

=> sin²θ + cos²θ = 1 ; Hence proved .

Further –

Dividing eq-(1) by p² , we get ;

=> (p² + b²)/p² = h²/p²

=> p²/p² + b²/p² = h²/p²

=> 1 + cot²θ = cosec²θ

=> cosec²θ - cot²θ = 1

Also ,

Dividing eq-(1) by b² , we get ;

=> (p² + b²)/b² = h²/b²

=> p²/b² + b²/b² = h²/b²

=> tan²θ + 1 = sec²θ

=> sec²θ - tan²θ = 1

Hence ;

• sin²θ + cos²θ = 1
• cosec²θ - cot²θ = 1
• sec²θ - tan²θ = 1

Answer 2

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Step-by-step explanation:

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