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Mathematics
Mathematics
Problem
For any acute angle
θ
\theta
θ
, define
H
(
θ
)
=
(
sec
2
θ
−
tan
2
θ
)
+
[
csc
2
(
90
∘
−
θ
)
−
cot
2
(
90
∘
−
θ
)
]
.
H(\theta)=\bigl(\sec^{2}\theta-\tan^{2}\theta\bigr)+\bigl[\csc^{2}(90^{\circ}-\theta)-\cot^{2}(90^{\circ}-\theta)\bigr].
H
(
θ
)
=
(
sec
2
θ
−
tan
2
θ
)
+
[
csc
2
(
9
0
∘
−
θ
)
−
cot
2
(
9
0
∘
−
θ
)
]
.
Calculate the value of
H
(
θ
)
H(\theta)
H
(
θ
)
when
θ
=
42
∘
\theta=42^{\circ}
θ
=
4
2
∘
without using a calculator, showing all working.
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