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Miike012
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Homework Statement
Prove:
cosA/(1-tanA) + sinA/(1-cotA) = sinA + cosA
I have tryed turning tan and cot into sin and cos and everything but I can not prove it... can some one help??
Thank you.
Miike012 said:cos/(1-sin/cos)+ sin/(1-cos/sin)
cos^2/(cos-sin) + sin^2/(sin-cos)
(cos^2 - Sin^2)/(cos-sin)
Miike012 said:cos/(1-sin/cos)+ sin/(1-cos/sin)
cos^2/(cos-sin) + sin^2/(sin-cos)
(cos^2 - Sin^2)/(cos-sin)
The purpose of proving this equation is to show that it is true for all values of A. This is important because it allows us to confidently use the equation in other mathematical calculations and proofs.
To prove this equation, we can use the fundamental trigonometric identities and algebraic manipulations. By substituting 1/tanA for cotA and 1/cotA for tanA, we can simplify the left side of the equation to (cosA + sinA) / (1 - tanA). Then, by using the Pythagorean identity sin^2A + cos^2A = 1, we can further simplify the left side to equal sinA + cosA, which is equal to the right side of the equation.
The fundamental trigonometric identities used in proving this equation are the Pythagorean identity (sin^2A + cos^2A = 1), the quotient identity (tanA = sinA/cosA), and the reciprocal identities (cotA = 1/tanA and cotA = cosA/sinA).
Yes, the equation cosA/(1-tanA) + sinA/(1-cotA) = sinA + cosA can be represented visually using a unit circle. By drawing a right triangle within the unit circle and labeling the sides with the appropriate trigonometric functions, we can see that both sides of the equation are equal to the length of the hypotenuse (represented by the radius of the unit circle).
The significance of proving this equation is that it strengthens our understanding and knowledge of trigonometric identities and their applications. It also allows us to confidently use this equation in other mathematical calculations and proofs. Additionally, proving this equation can serve as a basis for more complex mathematical concepts and theories.