Question

Simplify: (sec θ + tan θ) x (1 – sin θ)

Answer 1

cosθ

Explanation:

Given the expression (sec θ + tan θ) x (1 – sin θ), to simplify he expression the following steps must be followed. First we must know that from trigonometry identity, secθ = 1/cos θ and tanθ = sinθ/cosθ

Substituting this foremulas in the expression we will have:

(sec θ + tan θ) x (1 – sin θ)

= (1/cosθ + sinθ/cosθ)* (1-sinθ)

= (1+sinθ/cosθ) *1-sinθ

= {(1+sinθ)(1-sinθ)/cosθ}

= (1-sinθ+sinθ-sin²θ)/cosθ

= 1-sin²θ/cosθ

Also, from pythagoras therorem, sin²θ+cos²θ = 1

cos²θ = 1-sin²θ

substituting cos²θ = 1-sin²θ into the final expression above we have:

1-sin²θ/cosθ = cos²θ/cosθ

= cosθ

(sec θ + tan θ) x (1 – sin θ) = cosθ