Question

prove geometrically
trigonometric identities

$Sin²A+cos²A=1$
$1+tan²A=sec²A$
$Cot²A+1=cosec²A$
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Answer 1

$\huge\mathrm{Answer}$

1) To proof :-

cos²A + sin²A = 1

Prove :-

⭐ Refer for the triangle in the above attachment

According to the Pythagoras theorem we know,

c² = a² + b²

In triangle,

Base = b

Perpendicular = a

Hypotenuse = c

Dividing both side by c² we get,

c²/c² = a²/c² + b²/c²

sin²A = Perpendicular / hypotenuse

sin²A = a / c [ According to triangle ]

cos²A = Base / Hypotenuse

cos²A = b/c [ According to triangle ]

1 = ( a/c ) ² + ( b/c ) ²

➡ sin²A + cos²A = 1 ( Proved )

2) To proof :-

1 + tan²A = sec²A

Prove :-

1 + tan²A

= 1 + sin²A / cos²A

= ( cos²A + sin²A) / cos²A

= 1 / cos²A [ As proves above that, sin²A + cos²A = 1 ]

= sec² A ( Proved )

3)

To proof :-

1 + cot²A = cosec²A

Prove :-

cot²A + 1

= cos²A / sin²A + 1

= ( cos²A + sin²A ) / sin²A

= 1 / sin²A

= cosec² A ( Proved )