Question

Which of the following is a simpler form of the expression sin (theta)sec(theta)÷ cos(theta)tan(theta)

Answer 1

The simplified form of the given expression
`(\sin\theta \cdot \sec\theta)/(\cos\theta\cdot \tan\theta)` is
`\sec\theta`

Explanation:

Given: Expression
`(\sin\theta \cdot \sec\theta)/(\cos\theta\cdot \tan\theta)`

We have to writ the given expression in simplified form.

Consider , The given expression
`(\sin\theta \cdot \sec\theta)/(\cos\theta\cdot \tan\theta)`

Since, we know,

`\sec\theta=(1)/(\cos\theta)`

and
`\tan\theta=(\sin\theta)/(\cos\theta)`

Substitute, we have,

`(\sin\theta \cdot \sec\theta)/(\cos\theta\cdot \tan\theta)=\frac{\sin\theta \cdot(1)/(\cos\theta)} {\cos\theta\cdot (\sin\theta)/(\cos\theta) }`

Simplify, we have,

`=\frac{(1)/(\cos\theta)} {\cos\theta\cdot (1)/(\cos\theta) }`

Simplify further, we get,

`=(1)/(\cos\theta)=\sec\theta`

Thus, The simplified form of the given expression
`(\sin\theta \cdot \sec\theta)/(\cos\theta\cdot \tan\theta)` is
`\sec\theta`

Answer 2

Answer:
`sec \theta`.

Explanation: Given expression :
`(sin \theta \ sec \theta )/(cos \theta \ tan \theta) .`

We know,

`sec \theta = (1)/(cos \theta)`

`tan \theta = (sin \theta)/(cos \theta)`

Substituting those values in given expression, we get

`(sin \theta \ sec \theta )/(cos \theta \ tan \theta) .` =
`\frac{sin \theta\ (1)/(cos \theta) } {cos \theta\ (sin \theta)/(cos \theta) }`

=
`(sin \theta \ cos \theta)/(cos \theta \ cos \theta \ sin \theta)`

Crossing out
`sin \theta \ cos \theta` from top and bottom, we get

=
`(1)/(cos \theta )`

=
`sec \theta`.