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Question: How do you verify the identity \( 1 - \cos 2x = \tan x\sin 2x \) ?...

How do you verify the identity 1−cos⁡2x=tan⁡xsin⁡2x1 - \cos 2x = \tan x\sin 2x ?

Explanation

Solution

Hint : First we will evaluate the right-hand of the equation and then further the left-hand side of the equation. We will use the identity sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 . Then we will try to factorise and simplify the terms so that the left-hand side matches the right-hand side.

Complete step-by-step answer :
We will start off by solving the right-hand side of the equation. Here, we will be using the double angle identities.
sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2\sin x\cos x and we can write tan⁡x\tan x as  sin⁡xcos⁡x\,\dfrac{{\sin x}}{{\cos x}} .
Hence, the expression can be written as,
=tan⁡xsin⁡2x = sin⁡xcos⁡x×2sin⁡xcos⁡x   = \tan x\sin 2x \\\ = \,\dfrac{{\sin x}}{{\cos x}} \times 2\sin x\cos x \;
After we further simplify the expression it becomes,
2sin⁡2x2{\sin ^2}x
Now we apply the trigonometric identity, sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 .
Hence, the expression becomes,
=2sin⁡2x =2(1−cos⁡2x)   = 2{\sin ^2}x \\\ = 2(1 - {\cos ^2}x) \;
Now we will open the brackets and try to simplify the expression.
=2(1−cos⁡2x) =2−2cos⁡2x =−1(2cos⁡2x−2)   = 2(1 - {\cos ^2}x) \\\ = 2 - 2{\cos ^2}x \\\ = - 1(2{\cos ^2}x - 2) \;
As we know that cos⁡2x=2cos⁡2x−1\cos 2x = 2{\cos ^2}x - 1 and since we need 2cos⁡2x−12{\cos ^2}x - 1 to get cos⁡2x\cos 2x .
Therefore, we will rewrite the expression.
=−1(−1+2cos⁡2x−1) =−1(−1+cos⁡2x)   = - 1( - 1 + 2{\cos ^2}x - 1) \\\ = - 1( - 1 + \cos 2x) \;
Now finally the expression will become,
1−cos⁡2x1 - \cos 2x
Now we will check the left-hand side of the equation.
The expression in the left-hand side is 1−cos⁡2x1 - \cos 2x .
As we can see that the left-hand side equals the right-hand side.
Hence, we proved that 1−cos⁡2x=tan⁡xsin⁡2x1 - \cos 2x = \tan x\sin 2x .
So, the correct answer is “ 1−cos⁡2x=tan⁡xsin⁡2x1 - \cos 2x = \tan x\sin 2x ”.

Note : While choosing the side to solve, always choose the side where you can directly apply the trigonometric identities. Also, remember the trigonometric identities sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 and cos⁡2x=2cos⁡2x−1\cos 2x = 2{\cos ^2}x - 1 . While opening the brackets make sure you are opening the brackets properly with their respective signs. Also remember that tan⁡x= sin⁡xcos⁡x\tan x = \,\dfrac{{\sin x}}{{\cos x}} .
While applying the double angle identities, first choose the identity according to the terms you have then choose the terms from the expression involving which you are using the double angle identities.