Verify: $(cot x - csc x) (cos x + 1) = - sin x$ $(cotx - cscx)(cosx +1) =-sinx$ ?

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Answer 1 · 2016-08-01T01:28:43

Expand and simplify LHS - Apply identity ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x = 1$

To verify: $(cot x - csc x) (cos x + 1) = - sin x$ $(cotx - cscx)(cosx +1) =-sinx$

LHS= $(\frac{cos x}{sin x} - \frac{1}{sin x}) (cos x + 1)$ $(cosx/sinx - 1/sinx)(cosx+1)$

$= \frac{1}{sin x} (cos x - 1) (cos x + 1)$ $=1/sinx(cosx-1)(cosx+1)$

$= \frac{1}{sin x} ({cos}^{2} x - 1)$ $=1/sinx(cos^2x-1)$

Since ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x = 1$ $\to ({cos}^{2} x - 1) = - {sin}^{2} x$ $-> (cos^2x-1) = -sin^2x$

LHS = $- \frac{{sin}^{2} x}{sin x} = - sin x$ $-sin^2x/sinx = -sinx$ = RHS

Verify: (cotx−cscx)(cosx+1)=−sinx(cotx - cscx)(cosx +1) =-sinx ?