Question

1/cos x + tan x [identity] cos x / 1 - sin x​

Answer 1

The simplified expression is
`(sin\theta+cos\theta)/(1- sin^2 \theta)` .

To simplify the expression
`(1)/(cos \theta) + tan \theta = (cos \theta)/(1 - sin\theta)` , let's follow a similar approach as before:

Combine the fractions in the numerator:

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

Invert and multiply by the reciprocal of the denominator:

`(sin \theta + cos \theta)/(cos \theta) * (1 - sin\theta)/(cos\theta)`
`= ((sin\theta+cos\theta)(1- sin\theta))/(cos^2\theta)`

​Expand the numerator:

=
`(sin\theta+cos\theta- sin\theta cos\theta + cos\theta sin\theta)/(cos^2\theta)`

Combine like terms:

=
`(sin\theta+cos\theta)/(cos^2 \theta)`

​Now, if you want to express this in terms of trigonometric identities, you can use the Pythagorean identity
`sin^2 \theta +cos^2\theta = 1` to replace
`cos^2\theta`in the denominator:

=
`(sin\theta+cos\theta)/(1- sin^2 \theta)`

Therefore, the simplified expression is
`(sin\theta+cos\theta)/(1- sin^2 \theta)` .