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Use trigonometric identities to transform the left side of the equation into the right side (0 < theta < /2).
sinθ/cosθ + cosθ/sinθ = cscθsecθ
Full explanation
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To transform the left side of the equation
sin
cos
+
cos
sin
cosθ
sinθ
+
sinθ
cosθ
into
csc
sec
cscθsecθ, we'll use trigonometric identities step by step.
Start with the given expression:
sin
cos
+
cos
sin
cosθ
sinθ
+
sinθ
cosθ
Find a common denominator:
To combine these fractions, we need a common denominator. The common denominator here is
sin
cos
sinθcosθ.
sin
2
sin
cos
+
cos
2
sin
cos
sinθcosθ
sin
2
θ
+
sinθcosθ
cos
2
θ
Combine the fractions:
sin
2
+
cos
2
sin
cos
sinθcosθ
sin
2
θ+cos
2
θ
Apply the Pythagorean identity:
Recall that
sin
2
+
cos
2
=
1
sin
2
θ+cos
2
θ=1. Substitute this identity into the expression:
1
sin
cos
sinθcosθ
1
Express in terms of cosecant and secant:
Now, rewrite
1
sin
cos
sinθcosθ
1
using the identities:
1
sin
cos
=
csc
sec
sinθcosθ
1
=cscθsecθ
Therefore, we have shown that:
sin
cos
+
cos
sin
=
csc
sec
cosθ
sinθ
+
sinθ
cosθ
=cscθsecθ
This completes the transformation using trigonometric identities.:
Step-by-step explanation:
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