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  1. #1
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    Trigonometric identities

    Prove each identity using a t-chart.

    Left hand side Right hand side

    a) csc2 x(1 - cos2 x)=1

    b) (cot x + tan x)/sec x = csc x
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  2. #2
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    Re: Trigonometric identities

    Quote Originally Posted by alejandro View Post
    Prove each identity using a t-chart.

    Left hand side Right hand side

    a) csc2 x(1 - cos2 x)=1

    b) (cot x + tan x)/sec x = csc x
    ok ... have you made the requisite t-charts to verify each identity?
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  3. #3
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    Re: Trigonometric identities

    could you clarify as to what do you mean by " using t-charts". Does it mean pluging in values of trigonometric ratios in terms of the sides of a right triangle like, adjacent, opposite & hypotenuse etc??
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    Re: Trigonometric identities

    Hello, alejandro!

    I'm not familiar with a t-chart.


    \text{(a) Prove: }\:\csc^2x(1-\cos^2x) \:=\:1

    Since \csc x = \tfrac{1}{\sin x},\,\text{then: }\:\csc^2\!x  = \tfrac{1}{\sin^2\!x}

    Since \sin^2\!x + \cos^2\!x \:=\:1,\,\text{ then: }\:1 - \cos^2\!x \:=\:\sin^2\!x

    The left side becomes: . \tfrac{1}{\sin^2\!x}\cdot\sin^2\!x \;=\;1




    \text{(b) Prove: }\:\frac{\cot x + \tan x}{\sec x} \:=\:\csc x

    We have: . \frac{\dfrac{\cos x}{\sin x} + \dfrac{\sin x}{\cos x}}{\dfrac{1}{\cos x}}


    Multiply by \frac{\sin x\cos x}{\sin x\cos x}:

    . . \frac{\sin x\cos x\left(\dfrac{\cos x}{\sin x} + \dfrac{\sin x}{\cos x}\right)}{\sin x\cos x \left(\dfrac{1}{\cos x}\right)} \;=\;\frac{\overbrace{\cos^2\!x + \sin^2\!x}^{\text{This is 1}}}{\sin x} \;=\;\frac{1}{\sin x} \;=\;\csc x
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    Re: Trigonometric identities

    for exercise a)csc2 x(1 - cos2 x)=1 I have to prove csc2 x(1 - cos2 x)=1
    1 = 1
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    Re: Trigonometric identities

    Quote Originally Posted by alejandro View Post
    for exercise a)csc2 x(1 - cos2 x)=1 I have to prove csc2 x(1 - cos2 x)=1
    1 = 1
    Soroban completed that proof in post #4 ... did you not see it?
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    Re: Trigonometric identities

    Quote Originally Posted by alejandro View Post
    Prove each identity using a t-chart.

    Left hand side Right hand side

    a) csc2 x(1 - cos2 x)=1

    b) (cot x + tan x)/sec x = csc x
    We can also do it alternatively as shownTrigonometric identities-untitled-1.pngTrigonometric identities-untitled.png
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