Find the Value of cot(π/4 + x) cot(π/4 − x)

Solution

Using the identity:

\[ \cot\theta=\frac{1}{\tan\theta} \]

Therefore,

\[ \cot\left(\frac{\pi}{4}+x\right) \cot\left(\frac{\pi}{4}-x\right) = \frac{1}{ \tan\left(\frac{\pi}{4}+x\right) \tan\left(\frac{\pi}{4}-x\right) } \]

Now use:

\[ \tan\left(\frac{\pi}{4}+x\right) = \frac{1+\tan x}{1-\tan x} \]

and

\[ \tan\left(\frac{\pi}{4}-x\right) = \frac{1-\tan x}{1+\tan x} \]

Multiplying,

\[ \tan\left(\frac{\pi}{4}+x\right) \tan\left(\frac{\pi}{4}-x\right) =1 \]

Hence,

\[ \cot\left(\frac{\pi}{4}+x\right) \cot\left(\frac{\pi}{4}-x\right) = \frac{1}{1} =1 \]

Therefore,

\[ \boxed{1} \]