prove that secθ + tanθ = cos/(1-sinθ )

Solution

We can use the trigonometric identity:

secθ = 1/cosθ tanθ = sinθ/cosθ

Substituting these identities into the left-hand side of the equation, we get:

secθ + tanθ = 1/cosθ + sinθ/cosθ

Combining the two fractions, we get:

secθ + tanθ = (1 + sinθ)/cosθ

To obtain the right-hand side of the equation, we use the identity:

cos²θ = 1 – sin²θ

Rearranging this identity, we get:

cos²θ + sin²θ = 1

Dividing both sides by cos²θ, we get:

1 + tan²θ = sec²θ

Substituting this identity into the right-hand side of the equation, we get:

cos/(1-sinθ) = cosθ/((1-sinθ)cosθ) = 1/(1-sinθ/cosθ)

Using the identity tanθ = sinθ/cosθ, we get:

cos/(1-sinθ) = 1/(1-tanθ)

Substituting this into the previous expression, we get:

cos/(1-sinθ) = 1/(1-tanθ) = 1/(1-(sinθ/cosθ))

Simplifying this expression, we get:

cos/(1-sinθ) = cosθ/(cosθ-sinθ)

Therefore, we have:

secθ + tanθ = (1 + sinθ)/cosθ = cosθ/(cosθ-sinθ)

which is the same as:

secθ + tanθ = cos/(1-sinθ)

Hence, we have proved that secθ + tanθ = cos/(1-sinθ) using trigonometric identities.