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I am unable to prove this trigonometric identity

$$\frac{\tan (A)}{\tan (A)}+\frac{\cot (A)}{\cot (A)}=\frac{1}{1-2\cos^2(A)}$$

I have tried to transform the left-hand side and stuck with this

$$\frac{2\sin(A)\cos(A)}{\sin(A)\cos(A)}$$

And I have tried to transform the right-hand side by changing the $$2\cos^2(A)$$ to $$\frac{2}{\sec^2(A)}$$, and used the trigonometric identity $$1+\tan^2(A)=\sec^2(A)$$ and got this instead

$$\frac{1+\tan^2(A)}{\tan^2(A)-1}$$ which I can transform to $$\frac{\cot(A)+\tan(A)}{\tan(A)-\cot(A)}$$.

I cannot get both sides equal, help please?

bjcolby15
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user569622
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    I think there's a typo as the left side is always $1 + 1 = 2$, so it's independent of $A$, while the right side is definitely not a constant. – John Omielan Dec 30 '18 at 01:09
  • @JohnOmielan but that's exactly how the question on textbook is written? – user569622 Dec 30 '18 at 01:11
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    Then your textbook must have a typo, because this identity clearly can not be proven true. – Noble Mushtak Dec 30 '18 at 01:12
  • @JohnOmielan does that mean every equation that has a dependent variable on one side and a constant on the other side cannot be proven? – user569622 Dec 30 '18 at 01:14
  • @NobleMushtak I see, thank you for the answer. That must be the case, the book is so old – user569622 Dec 30 '18 at 01:15
  • @user569622 No, it just means that the dependent variable usually cannot be fully variable but, instead, must have a value so the value of its side of the equation matches. This usually limits the possible set of values to either a small finite # (e.g., 0, 1 or 2) or a specific set (e.g., for trigonometric functions, $+ 2k\pi$, for all integral values of $k$). – John Omielan Dec 30 '18 at 01:17
  • You check whether a given identity is false by plotting each side of the identity as separate functions, e.g. on desmos.com. if the graphs don't match up then you know something must be wrong. – pshmath0 Dec 30 '18 at 02:15
  • @Antinous oh, i see. that is useful, thank you – user569622 Dec 30 '18 at 02:38

1 Answers1

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One way we can prove the identity false is as follows:

$$\begin {align} \dfrac {\tan A} {\tan A} + \dfrac{\cot A}{\cot A} = \dfrac {1}{1-2\cos^2 2A} \\ 2 = \dfrac {1}{1-2\cos^2 2A} \\ 2 (1-2\cos^2 2A) = 1 \\ 2 - 4\cos^2 2A = 1 \\ - 4\cos^2 2A = \dfrac {1}{2} \\ \cos^2 2A = -\dfrac {1}{8} \end {align}$$

Since the last line would require us to take the square root of a negative number, $A$ does not exist, and the identity is false.

bjcolby15
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