How do you verify $sin x + cos x * cot x = csc x$ ?

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Answer 1 · 2016-03-02T01:43:37

Recall the following reciprocal, tangent, and Pythagorean identities:

$1$ . $color(orange)cotx=1/tanx$

$2$ . $color(blue)cscx=1/sinx$

$3$ . $color(purple)tanx=sinx/cosx$

$4$ . $color(brown)(sin^2x+cos^2x)=1$

Proving the Identity
$1$ . Simplify by rewriting the left side of the identity in terms of $sinx$ and $cosx$ .

$sinx+cosx*color(orange)cotx=color(blue)cscx$

Left side:

$sinx+cosx*1/color(purple)tanx$

$=sinx+cosx*1/(sinx/cosx)$

$=sinx+cosx*(1-:sinx/cosx)$

$=sinx+cosx*(1/1*cosx/sinx)$

$=sinx+cosx*(cosx/sinx)$

$2$ . Simplify by multiplying $cosx$ by $cosx/sinx$ .

$=sinx+cos^2x/sinx$

$3$ . Find the L.C.M. (lowest common multiple) to rewrite the left hand side as a fraction.

$=(sinx(sinx)+cos^2x)/sinx$

$4$ . Simplify.

$=color(brown)((sin^2x+cos^2x))/sinx$

$=1/sinx$

$=color(green)(cscx)$

$:.$ , left side $=$ right side.

How do you verify sin x + cos x * cot x = csc xsin x + cos x * cot x = csc x?