prove : 1-tan^2 theta/cos ^2 theta-1= tan^2theta
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Step-by-step explanation:
We seek to prove that:
1
−
tan
2
θ
1
+
tan
2
θ
≡
cos
(
2
θ
)
We will require the following trigonometric definitions/identities:
tan
ϕ
=
sin
ϕ
cos
ϕ
sin
2
ϕ
+
cos
ϕ
≡
1
cos
2
ϕ
≡
cos
2
ϕ
−
sin
2
ϕ
Consider the LHS of the given expression:
L
H
S
=
1
−
tan
2
θ
1
+
tan
2
Theta
=
1
−
sin
2
θ
cos
2
θ
1
+
sin
2
θ
cos
2
θ
=
cos
2
θ
−
sin
2
θ
cos
2
θ
cos
2
θ
+
sin
2
θ
cos
2
θ
=
cos
2
θ
−
sin
2
θ
cos
2
θ
+
sin
2
θ
=
cos
2
θ
1
=
cos
2
θ
=
R
H
S
QED
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