Prove that sin theta/tan theta=cos theta/sec theta=0
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Precalculus Examples
Popular Problems Precalculus Verify the Identity sin(theta)tan(theta)+cos(theta)=sec(theta)
sin
(
θ
)
tan
(
θ
)
+
cos
(
θ
)
=
sec
(
θ
)
Start on the left side.
sin
(
θ
)
tan
(
θ
)
+
cos
(
θ
)
Write
tan
(
θ
)
in sines and cosines using the quotient identity.
sin
(
θ
)
sin
(
θ
)
cos
(
θ
)
+
cos
(
θ
)
Write
sin
(
θ
)
as a fraction with denominator
1
.
sin
(
θ
)
1
⋅
sin
(
θ
)
cos
(
θ
)
+
cos
(
θ
)
Combine.
sin
(
θ
)
sin
(
θ
)
1
cos
(
θ
)
+
cos
(
θ
)
Multiply
sin
(
θ
)
sin
(
θ
)
.
sin
2
(
θ
)
1
cos
(
θ
)
+
cos
(
θ
)
Multiply
cos
(
θ
)
by
1
.
sin
2
(
θ
)
cos
(
θ
)
+
cos
(
θ
)
Apply Pythagorean identity in reverse.
1
−
cos
2
(
θ
)
cos
(
θ
)
+
cos
(
θ
)
Simplify.
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1
cos
(
θ
)
Rewrite
1
cos
(
θ
)
as
sec
(
θ
)
.
sec
(
θ
)
Because the two sides have been shown to be equivalent, the equation is an identity.
sin
(
θ
)
tan
(
θ
)
+
cos
(
θ
)
=
sec
(
θ
)
is an identity
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