Math, asked by lugvv, 1 year ago

(cos/sin+1/sin)²=1+cos/1-cos

Answers

Answered by roseannramos6
0

Modifying just the left-hand side:


We can use the Pythagorean Identity to rewrite

sin

2

x

. The Pythagorean Identity states that


¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

sin

2

x

+

cos

2

x

=

1

a

a

−−−−−−−−−−−−−−−−−−−−−


Which can be rearranged to say that


¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

sin

2

x

=

1

cos

2

x

a

a

−−−−−−−−−−−−−−−−−−−−−


Thus, we see that


sin

2

x

1

cos

x

=

1

cos

2

x

1

cos

x


We can now factor the numerator of this fraction.

1

cos

2

x

is a difference of squares, which can be factored as:


¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

a

2

b

2

=

(

a

+

b

)

(

a

b

)

a

a

−−−−−−−−−−−−−−−−−−−−−−−−−−


This can be applied to

1

cos

2

x

as follows:


¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

1

cos

2

x

=

1

2

(

cos

x

)

2

=

(

1

+

cos

x

)

(

1

cos

x

)

a

a

−−−−−−−−−−−−−−−−−−−−−−−−−


Therefore,


1

cos

2

x

1

cos

x

=

(

1

+

cos

x

)

(

1

cos

x

)

1

cos

x


Since there is

1

cos

x

present in both the numerator and denominator, it can be cancelled:


(

1

+

cos

x

)

(

1

cos

x

)

1

cos

x

=

(

1

+

cos

x

)

(

1

cos

x

)

(

1

cos

x

)

=

1

+

cos

x


This is what we initially set out to prove.

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