How do you simplify 1+cos(x)/1+csc(x)?

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Answer 1 · 2017-10-31T10:58:54

$(1+cosx)/(1+cscx) = (2sinx+sin2x)/(2(sinx+1))$

Start by splitting the numerator and denominator and simplifying.

$1 + cscx = 1 + 1/sinx$
$=(sinx +1)/sinx$

Now perform the division (multiply the numerator by the reciprocal denominator):

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

This fraction becomes: $(sinx + sinx cosx)/(sinx +1)$

By the double angle formula for $sin$ , $sinxcosx=1/2sin2x$ .

Bringing out the factor of $1/2$ in the numerator, the fraction now becomes: $(1/2(2sinx +sin2x))/(sinx +1)$

Simplify to find the fraction: $(2sinx + sin2x)/(2(sinx+1))$ .