Verified answer

Rewrite the LHS as a single fraction.

[sin^2 + (1 + cos)^2] / [sin(1 + cos)] =

[sin^2 + 1 + 2cos + cos^2] / [sin(1 + cos)] =

Since, sin^2 + cos^2 = 1, the numerator is

[2 + 2cos] / [sin(1 + cos)] =

[2(1 + cos)] / [sin(1 + cos)] =

2 / sin = 2csc.

L.H.S = (sinθ)/(1+cosθ) + (1+cosθ)/(sinθ)

= [sinθ * sinθ + (1+cosθ)(1+cosθ)]/[sinθ (1+cosθ)]

= [sin^2θ+ 1+ 2 cosθ + cos^2θ]/[sinθ (1+cosθ)]

= [ 2 + 2 cosθ]/[sinθ(1+cosθ)]

= [2(1+cosθ)]/[sinθ(1+cosθ)]

= 2/sinθ = 2 cscθ = R.H.S ...................(Q.E.D)

sin?/(a million-cos?)) + ((a million-cos?)/sin?) =2csc? Multiply the 1st fraction by applying (a million+ cos?) / (a million + cos?) and the 2d by applying sin? / sin?. (sin?(a million+ cos?)) / ((a million-cos?)(a million+cos?))+ ((sin?(a million-cos?))/sin²?) remember: sin²? = a million - cos²?, and that (a million-cos?)(a million+cos?) = a million - cos²? (sin?(a million+cos?)sin?(a million+cos?) + sin? (a million-cos?)] / sin²? [sin? + sin?cos? + sin? - sin?cos? ((sin?(a million-cos?))/sin²?) because of the fact the fractions have elementary denminators combine them: [sin?(a million+cos?) + sin? (a million-cos?)] / sin²? [sin? + sin?cos? + sin? - sin?cos?] / sin²? 2sin?/sin²? 2/sin? finally sin? =a million/csc? consequently 2/sin? = 2csc? consequently, 2csc? = 2csc?