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4 votes
(sec x + csc x)(cos x − sin x) = cot x − tan x

Verify the answer

User Wei An
by
5.9k points

2 Answers

5 votes
Step-by-step explanation:
The reciprocal identities
csc
θ
=
1
sin
θ
sec
θ
=
1
cos
θ
cot
θ
=
1
tan
θ
The quotient identities:
tan
θ
=
sin
θ
cos
θ
cot
θ
=
cos
θ
sin
θ
Applying all these identities, on both sides, we get:
1
sin
x
+
1
cos
x
sin
x
+
cos
x
=
cos
x
sin
x
+
sin
x
cos
x
cos
x
+
sin
x
cos
x
sin
x
sin
x
+
cos
x
=
cos
x
sin
x
+
sin
x
cos
x
1
sin
x
+
cos
x
×
cos
x
+
sin
x
cos
x
sin
x
=
cos
x
sin
x
+
sin
x
cos
x
1
cos
x
sin
x
=
cos
2
x
+
sin
2
x
sin
x
cos
x
Applying the pythagorean identity
sin
2
x
+
cos
2
x
=
1
on the right side, we get:
1
cos
x
sin
x
=
1
sin
x
cos
x

Applying the pythagorean identity
sin
2
x
+
cos
2
x
=
1
on the right side, we get:
1
cos
x
sin
x
=
1
sin
x
cos
x
User Kevin Smyth
by
5.4k points
3 votes

Answer:

Explanation:

The reciprocal identities

csc

θ

=

1

sin

θ

sec

θ

=

1

cos

θ

cot

θ

=

1

tan

θ

The quotient identities:

tan

θ

=

sin

θ

cos

θ

cot

θ

=

cos

θ

sin

θ

Applying all these identities, on both sides, we get:

1

sin

x

+

1

cos

x

sin

x

+

cos

x

=

cos

x

sin

x

+

sin

x

cos

x

cos

x

+

sin

x

cos

x

sin

x

sin

x

+

cos

x

=

cos

x

sin

x

+

sin

x

cos

x

1

sin

x

+

cos

x

×

cos

x

+

sin

x

cos

x

sin

x

=

cos

x

sin

x

+

sin

x

cos

x

1

cos

x

sin

x

=

cos

2

x

+

sin

2

x

sin

x

cos

x

Applying the pythagorean identity

sin

2

x

+

cos

2

x

=

1

on the right side, we get:

1

cos

x

sin

x

=

1

sin

x

cos

x

Hopefully this helps!

User Avikam
by
4.8k points
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