Question

Describe, in your own words, the identity sin^2(x) + cos^2(x) = 1

Answer 1

Step-by-step explanation:

${ \sin }^{2} x + { \cos}^{2} x = 1 \\ \sin(x) = \frac{opp}{hyp} \: \: \: \: \: \: \: \: \: \: { \sin }^{2}x = ( { \frac{opp}{hyp}) }^{2} \\ \cos(x) = \frac{adj}{hyp} \: \: \: \: \: \: \: \: \: { \cos }^{2} x = {( \frac{adj}{hyp} )}^{2} \\ = \frac{ {opp}^{2} }{ {hyp}^{2} } + \frac{ {adj}^{2} }{ {hyp}^{2} } = 1\\ = \frac{ {opp}^{2} + {adj}^{2} }{ {hyp}^{2} } = 1 \\ according \: to \: pythagoras \: theorem\\ {opp}^{2} + {adj}^{2} = {hyp}^{2} \\ \\ hence \: proved$

hope you find your answer

Answer 2

Explanation:

From the Pythagoras theorem we know that

a²+b² = c²

Now let us express  a and b by using sine and cosine

thus,

$\sin x = \frac{b}{c}\\\\b = c\sin x...(1)\\\\\cos x = \frac{a}{c}\\\\a= c \cos x..(2)\\\\\text{use 1 and 2 in PGT}\\\\(c\cos x)^2+(c\sin x)^2= c^2\\\\c^2\cos^2x+c^2\sin^2x = c^2\\\\\text{divide the equation by c square we get}\\\\\cos^2x +\sin^2x = 1$

hence proved

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