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Answer 1 · 2017-07-23T13:37:58

False.

We have: #(1 + frac(1)(cos(x)) + tan(x))(1 - cos(x) + tan(x)) = 2 tan(x)#

Let's check whether the two sides hold for #x = frac(pi)(4)#:

#Rightarrow (1 + frac(1)(cos(frac(pi)(4))) + tan(frac(pi)(4)))(1 - cos(frac(pi)(4)) + tan(frac(pi)(4))) = 2 tan(frac(pi)(4))#

#Rightarrow (1 + frac(1)(frac(sqrt(2))(2)) + 1)(1 - frac(sqrt(2))(2) + 1) = 2 cdot 1#

#Rightarrow (1 - frac(2)(sqrt(2)))(2 - frac(sqrt(2))(2)) = 2#

#Rightarrow (1 - frac(2 sqrt(2))(2))(frac(4 - sqrt(2))(2)) = 2#

#Rightarrow (1 - sqrt(2))(frac(4 - sqrt(2))(2)) = 2#

#Rightarrow frac(4 - sqrt(2) - 4 sqrt(2) + 2)(2) = 2#

#Rightarrow frac(6 - 5 sqrt(2))(2) ne 2#

Therefore, the statement is false.