Question

Cos x = tan X 1+ tan х2​

Answer 1

Coc x=tan x 1+ tan x2=cos x3

Answer 2

To solve the equation $\cos(x) = \frac{\tan(x)}{1 + \tan^2(x/2)}$, you can start by using trigonometric identities to simplify it. One useful identity in this case is:

$\tan(x) = \frac{2\tan(x/2)}{1 - \tan^2(x/2)}$.

Now, let's rewrite the equation with this identity:

$\cos(x) = \frac{\frac{2\tan(x/2)}{1 - \tan^2(x/2)}}{1 + \left(\frac{2\tan(x/2)}{1 - \tan^2(x/2)}\right)^2}$.

Simplify the equation further:

$\cos(x) = \frac{2\tan(x/2)}{1 - \tan^2(x/2)} \cdot \frac{1 - \tan^2(x/2)}{1 + \left(\frac{2\tan(x/2)}{1 - \tan^2(x/2)}\right)^2}$.

Now, simplify the equation by canceling common terms:

$\cos(x) = 2\tan(x/2) \cdot \frac{1 - \tan^2(x/2)}{1 + \frac{4\tan^2(x/2)}{(1 - \tan^2(x/2))^2}}$.

Next, simplify the denominator:

$\cos(x) = 2\tan(x/2) \cdot \frac{1 - \tan^2(x/2)}{1 + \frac{4\tan^2(x/2)}{1 - 2\tan^2(x/2) + \tan^4(x/2)}}$.

Now, let $u = \tan(x/2)$, so the equation becomes:

$\cos(x) = 2u \cdot \frac{1 - u^2}{1 + \frac{4u^2}{1 - 2u^2 + u^4}}$.

To simplify further, let's find a common denominator:

$\cos(x) = 2u \cdot \frac{(1 - u^2)(1 - 2u^2 + u^4)}{1 + 4u^2}$.

Expand the numerator:

$\cos(x) = 2u \cdot \frac{1 - 2u^2 + u^4 - u^2 + 2u^4 - u^6}{1 + 4u^2}$.

Combine like terms in the numerator:

$\cos(x) = 2u \cdot \frac{1 - 3u^2 + 3u^4 - u^6}{1 + 4u^2}$.

Now, you can cancel out common factors if they exist:

$\cos(x) = 2u \cdot \frac{(1 - u^2)^3}{(1 + 2u)(1 + 2u)}$.

$\cos(x) = 2u \cdot \frac{(1 - u^2)^3}{(1 + 2u)^2}$.

Finally, replace $u$