Question

Prove That (sec$-cosec$)/sec$+cosec$= (tan$-1)/tan$+1

Answer 1

Interpret sec, cosec, tan as functions of an angle θ. Start with the left-hand side:

(sec θ − cosec θ)/(sec θ + cosec θ)
= ((1/cos θ) − (1/sin θ))/((1/cos θ) + (1/sin θ))

Multiply numerator and denominator by sin θ cos θ to clear fractions:

= (sin θ − cos θ)/(sin θ + cos θ).

Now the right-hand side:

(tan θ − 1)/(tan θ + 1)
= ((sin θ/cos θ) − 1)/((sin θ/cos θ) + 1)
= ((sin θ − cos θ)/cos θ)/((sin θ + cos θ)/cos θ)
= (sin θ − cos θ)/(sin θ + cos θ).

Thus both sides equal (sin θ − cos θ)/(sin θ + cos θ), so the identity holds.

Domain: the equality is valid for angles θ for which sin θ ≠ 0, cos θ ≠ 0 and sin θ + cos θ ≠ 0 (equivalently tan θ and sec θ, cosec θ defined and denominators nonzero).