need help proving identity?

θ has been left out for greater clarity in reading

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LHS = sin / (1+cos) + (1+cos) / sin

put LHS under a common denominator

[sin² + (1+cos)² ] / (1+cos)•sin

(sin² + 1 + cos² + 2cos) / (1+cos)•sin

2(1+cos) / (1+cos)•sin

2/sin

multiply top & bottom by cos

2cos/(sin•cos)

2cot•sec

QED

from LHS

sin x / (1 + cos x) + (1 + cos x) / sin x

=(sin^2 x + 1 + 2 cos x + cos^2 x ) / (sin x)( 1 + cos x)

=( 2 + 2 cos x ) / (sin x)( 1 + cos x)

= 2 / sin x

= 2 cos x / (sin x cos x)

= 2cot x sec x

for more info check out http://www.a-maths-tuition.com/2010/02/proving-tri...

we will work the LHS going for a common denominator of sinx*(1+cosx) .

[The restrictions are that the cos x does not equal -1, the cos x does not equal 0, the sinx does not equal zero]

[sin^2 (x) +1+2cosx + cos^2 (x) ]/ sinx(1+cosx)

[2 + 2cosx] / sinx* (1+cosx)

2(1 + cosx) / sinx (1+cosx)

2/ sinx

==>multiply by cosx/cosx

2cosx/sinx cosx

===> split apart into the product of 3 terms

2 * cosx/sinx * 1/ cosx

2 * cotx * sec x = 2 *cotx* secx