Question

if sinx + cosx=a then show that (sinx-cosx) =√2-a(square)

Answer 1

sinx + cosx = a

squaring both sides

sin²x + cos²x +2sinxcosx = a²

2sinxcosx = a² - 1

(sinx - cosx )² = sin²x + cos²x - 2sinxcosx = 1 - (a² - 1) = 2-a²

sinx-cosx = √2-a²

hence proved.

Answer 2

$\sin x-\cos x=\sqrt{2-a^{2}}$, shown.

Step-by-step explanation:

We have,

$\sin x+\cos x=a$                  .........(1)

Let $\sin x-\cos x=x$            .........(2)

Show that, $\sin x-\cos x=\sqrt{2-a^{2}}$.

Squaring and adding equations (1) and (2), we get

$(\sin x+\cos x)^2+(\sin x-\cos x)^2=a^2+x^2$

⇒ $\sin^2 x+\cos^2 x+2\sin x\cos x+\sin^2 x+\cos^2 x-2\sin x\cos x=a^2+x^2$

⇒ $\sin^2 x+\cos^2 x+\sin^2 x+\cos^2 x=a^2+x^2$

⇒ $2\sin^2 x+2\cos^2 x=a^2+x^2$

⇒ $2(\sin^2 x+\cos^2 x)=a^2+x^2$

⇒ $2(1)=a^2+x^2$

Using the trigonometric identity,

$\sin^2 A+\cos^2 A=1$

⇒ $2=a^2+x^2$

⇒ $x^2=2-a^2$

⇒ $x=\sqrt{2-a^2}$

∴ $\sin x-\cos x=\sqrt{2-a^{2}}$, shown.