Prove trig identity (cot x -1)/(cot x +1)=(1-sin 2x)/(cos 2x)

• MHB
• karush
In summary, the given equations show that the numerator of the expression is equal to the numerator of the given equation, which is 1 minus the sine of 2x, while the denominator is equal to the cosine of 2x. This can be simplified to 1 minus the sine of 2x over the cosine of 2x, which is the same as the given equation.
karush
Gold Member
MHB
\begin{align*} \frac{\cot {x}-1}{\cot{x}+1}&=\frac{1-\sin 2x}{\cos 2x}\\ \frac{\cos {x}-\sin x}{\cos{x}+\sin x} \frac{\cos x-\sin x}{\cos x-\sin x}&= \\ \frac{\cos^2x-2\sin x\cos x+\cos^2 x}{\cos^2 x-\sin^2 x} \end{align*}

so far..

karush said:
\begin{align*} \frac{\cot {x}-1}{\cot{x}+1}&=\frac{1-\sin 2x}{\cos 2x}\\ \frac{\cos {x}-\sin x}{\cos{x}+\sin x} \frac{\cos x-\sin x}{\cos x-\sin x}&= \\ \frac{\cos^2x-2\sin x\cos x+\cos^2 x}{\cos^2 x-\sin^2 x} \end{align*}

so far..
Check your numerator. It should be $$\displaystyle cos^2(x) - 2~sin(x)~cos(x) + sin^2(x)$$.

Otherwise it's good. :)

-Dan

\begin{align*} \frac{\cot {x}-1}{\cot{x}+1}&=\frac{1-\sin 2x}{\cos 2x}\\ \frac{\cos {x}-\sin x}{\cos{x}+\sin x} \frac{\cos x-\sin x}{\cos x-\sin x}&= \\ \frac{\cos^2x-2\sin x\cos x+\sin^2 x} {\displaystyle cos^2x- sin^2x}=\\ \frac{1-\sin 2x}{\cos 2x} \end{align*}

hopefully

Yup, you got it. (Yes)

-Dan

1. What is a trig identity?

A trig identity is an equation that is true for all values of the variables involved. They are used to simplify complex trigonometric expressions and solve equations.

2. How do you prove a trig identity?

To prove a trig identity, you must manipulate one side of the equation until it is equivalent to the other side. This can be done by using trigonometric identities, properties, and algebraic manipulations.

3. What is the purpose of proving a trig identity?

Proving a trig identity helps to simplify complex expressions and solve equations involving trigonometric functions. It also helps to understand the relationships between different trigonometric functions.

4. What are the common trig identities used in proving equations?

Some common trig identities used in proving equations include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities.

5. How do you prove the given trig identity, (cot x -1)/(cot x +1)=(1-sin 2x)/(cos 2x)?

To prove this trig identity, we can start by manipulating the left side of the equation. We can use the identity cot x = cos x/sin x to rewrite the numerator and denominator. Then, we can use the Pythagorean identity sin^2 x + cos^2 x = 1 to simplify the expression. Finally, we can use the double angle identity sin 2x = 2sin x cos x to rewrite the right side of the equation and show that it is equivalent to the left side.

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