Question

sin^2A+cos^2A = 1 proved

Answer 1

Pythagoras theorem,

Hypotenuse² = base² + height² ----: ( 1 )







$\boxed{ \bold{ \: \sin( \theta) = \frac{hieght}{hypotenuse} \: \: \: \: \: \: \: \: \: \: \: and \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cos( \theta) = \frac{base}{hypotenuse} }}$


So,

$\boxed{ \bold{ \sin ^{2} ( \theta) = \frac{hieght ^{2} }{ {hypotenuse}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: and \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cos ^{2} ( \theta) = \frac{ {base}^{2} }{ {hypotenuse}^{2} } }}$







Now,



sin²A + cos²A


$\frac{ {height}^{2} }{hypotenuse ^{2} } + \frac{ {base}^{2} }{ {hypotenuse}^{2} } \\ \\ \\ \\ \frac{ {height}^{2} + {base}^{2} }{ {hypotenuse}^{2} }$





From ( i ), putting value.



$\frac{ {hypotenuse}^{2} }{ {hypotenuse}^{2} } \\ \\ \\ 1$



1 = 1

LHS = RHS




Hence, proved.

Answer 2

Holla ^_^


☸ Required Answer is :


✔⭕ As we know , Sin A = Perpendicular / Hypotenuse

Therefore , Sin²A = P²/H²

Similarly,

⭕✔ Cos A = Base / Hypotenuse

Then , Cos²A = B²/H²


A.T.Q

From, phytagoras theorem
H² = P² + B²

⏯ Solving given equation

▶ Sin²A + Cos² A = 1

L.H.S

▶ P²/ H² + B²/H²

By Taking LCM

▶ P²+B²/ H²

▶ H²/H² ( bcz H² = P²+B² )

▶ 1

L.H.S = R.H.S

Hence Proved.


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