How to solve this identity? Thank you!

Question

$1/sina + 1/tana = cot(a/2)$

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Answer 1 · 2017-12-10T12:42:52

If we know our trigonometric inverses right, we can simplify as follows:
$1/sin(a)=csc(a)$
$1/tan(a)=cot(a)$

$1/sin(a)+1/tan(a)=csc(a)+cot(a)$

This is just the result of the half angle formula for $cot$ :
$cot(theta/2)=csc(theta)+cot(theta)$

So we can simplify to:
$=cot(a/2)$ , which is what we wanted to prove.

Answer 2 · 2017-12-10T12:53:50

See explanation.

$L=1/sinalpha+1/tanalpha=1/sinalpha+cosalpha/sinalpha=$

$=(1+cosalpha)/sinalpha=(1+cos2*(alpha/2))/sin(2*alpha/2)=$

$(1+2cos^2(alpha/2)-1)/(2sin(alpha/2)cos(alpha/2))=$

$(2cos^2(alpha/2))/(2sin(alpha/2)cos(alpha/2))=$

$(cos^(cancel(2)}(alpha/2))/(sin(alpha/2)cancel(cos(alpha/2)))=$

$(cos(alpha/2))/(sin(alpha/2))=cot(alpha/2)=R$