If tanx = (sinα − cosα)/(sinα + cosα), Show that sinα + cosα = √2 cosx
Question
If
\[ \tan x = \frac{\sin\alpha-\cos\alpha} {\sin\alpha+\cos\alpha} \]
show that:
\[ \sin\alpha+\cos\alpha = \sqrt2\cos x \]
Proof
Given,
\[ \tan x = \frac{\sin\alpha-\cos\alpha} {\sin\alpha+\cos\alpha} \]
Using
\[ \sin\alpha-\cos\alpha = \sqrt2\sin\left(\alpha-\frac{\pi}{4}\right) \]
and
\[ \sin\alpha+\cos\alpha = \sqrt2\cos\left(\alpha-\frac{\pi}{4}\right) \]
\[ \tan x = \frac{ \sqrt2\sin\left(\alpha-\frac{\pi}{4}\right) }{ \sqrt2\cos\left(\alpha-\frac{\pi}{4}\right) } \]
\[ = \tan\left(\alpha-\frac{\pi}{4}\right) \]
Therefore,
\[ x=\alpha-\frac{\pi}{4} \]
\[ \alpha=x+\frac{\pi}{4} \]
Now,
\[ \sin\alpha+\cos\alpha = \sqrt2\cos\left(\alpha-\frac{\pi}{4}\right) \]
\[ = \sqrt2\cos x \]
Hence proved.