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Solve for [tex]\( x \)[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]

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Simplify the expression.

[tex]\[ \frac{\operatorname{sen} \theta}{\csc \theta}+\frac{\cos \theta}{\sec \theta}=1 \][/tex]


Sagot :

Absolutely, let’s simplify the given expression step-by-step to see how it equals 1.

Given expression:
[tex]\[ \frac{\sin \theta}{\csc \theta} + \frac{\cos \theta}{\sec \theta} \][/tex]

First, recall the definitions of cosecant ([tex]\(\csc\)[/tex]) and secant ([tex]\(\sec\)[/tex]) in terms of sine ([tex]\(\sin\)[/tex]) and cosine ([tex]\(\cos\)[/tex]):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]

Substituting these into the given expression:

[tex]\[ \frac{\sin \theta}{\frac{1}{\sin \theta}} + \frac{\cos \theta}{\frac{1}{\cos \theta}} \][/tex]

Now simplify each fraction:
[tex]\[ \frac{\sin \theta}{\frac{1}{\sin \theta}} = \sin \theta \cdot \sin \theta = \sin^2 \theta \][/tex]
[tex]\[ \frac{\cos \theta}{\frac{1}{\cos \theta}} = \cos \theta \cdot \cos \theta = \cos^2 \theta \][/tex]

Now substitute these back into the expression:
[tex]\[ \sin^2 \theta + \cos^2 \theta \][/tex]

Using the Pythagorean identity, which states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]

Thus, the simplified expression is:
[tex]\[ 1 \][/tex]

Therefore,
[tex]\[ \frac{\sin \theta}{\csc \theta} + \frac{\cos \theta}{\sec \theta} = 1 \][/tex]

And we’ve shown that the original expression simplifies to 1.