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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


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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: