a) You need to bring the terms to a common denominator such that:
`(cos x*(1+sin x) - cos x*(1 - sin x))/((1+sin x)(1 - sin x)) = 2 tan x`
You need to open the brackets such that:
`(cos x + cos x*sin x - cos x + cos x*...
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a) You need to bring the terms to a common denominator such that:
`(cos x*(1+sin x) - cos x*(1 - sin x))/((1+sin x)(1 - sin x)) = 2 tan x`
You need to open the brackets such that:
`(cos x + cos x*sin x - cos x + cos x* sin x)/(1 - sin^2 x) = 2 tan x`
Reducing like terms yields:
`(2cos x*sin x)/(1 - sin^2 x) = 2 tan x`
You need to remember that `1 - sin^2 x = cos^2 x` such that:
`(2cos x*sin x)/(cos^2 x) = 2 tan x`
Reducing by `cos x ` yields:
`(2sin x)/(cos x) = 2 tan x`
You need to substitute `sin x/cos x` by tan x such that:
`2 tan x = 2 tan x`
Hence, using formulas of trigonometry and the laws applied to difference of fractions that have different denominators yields that `cosx/(1-sinx) - cosx/(1+sinx) =2 tanx` .
b) You need to multiply by `cos x(1 + cos x)` both sides such that:
`(1 - sin x)(1 + cos x) = sin x*cos x`
You need to open the brackets such that:
`1 + cos x - sin x - sin x*cos x = sin x*cos x`
`1 + cos x - sin x = 2 sin x*cos x`
`1 + cos x - sin x = sin 2x`
Hence, making transformations to the left side yields that the expression `(1-sinx)/cosx != sinx/(1+cosx).`