Solving Pythagorean Identities - Trigonometry Calculator

Solving Pythagorean Identities


The Pythagorean identities in trigonometry are the three identities that come from the Pythagorean theorem. Recall that the Pythagorean theorem states that the hypotenuse squared of a right triangle is the sum of the square of each of the other two sides. a2 + b2 = c2 Where c stands for the hypotenuse, and a and b are other two sides of the right triangle. From this theorem, three identities can be determined from substituting in sine and cosine as follows:
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = cosec2 θ
 
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Result
Proof of Pythagorean Identities :
Tri-prism Lets drow an unit circle as showing in picture and draw an angle θ since it is a unit circle so line CP = 1, let draw the perpendicual lines to x and y axis as PN and PM.
As we know sinθ = opposite-side(O) / hypotunse and cosθ = adjacent-side (A) / hypotunuse
since hypotunuse = radius = 1 herefore we can write as follows:
sinθ = O and cosθ = A
We know the Pythagorean theorem as CP2 = PN2 + CN2 or 12 = O2 + A2
Now replace the value of O and A with sinθ and cosθ; so we will get
12 = (sinθ)2 + (cosθ)2
1 = sin2θ + cos2θ
sin2θ + cos2θ = 1 --------(1)
To prove other two indentities we can use following formulas:
           1 / sinθ = cscθ
           1 / cosθ = secθ
           sinθ / cosθ = tanθ
           cosθ / sinθ = cotθ
let take the basic indentity 1 = sin2θ + cos2θ and device both side by cos2θ
1 / cos2θ = sin2θ / cos2θ + cos2θ / cos2θ
1 / cos2θ = sin2θ / cos2θ + 1
apply basic formulas for 1 / cosθ = secθ and sinθ / cosθ = tanθ, then we will get
sec2θ = tan2θ + 1
1 + tan2θ = sec2θ -------------(2)
again take the basic indentity 1 = sin2θ + cos2θ and device both side by sin2θ
1 / sin2θ = sin2θ / sin2θ + cos2θ / sin2θ
1 / sin2θ = 1 + cos2θ / sin2θ
apply basic formulas for 1 / sinθ = cscθ and cosθ / sinθ = cotθ, then we will get
csc2θ = 1 + cot2θ
1 + cot2θ = csc2θ -------------(3)