Prove the identity secθ-tanθ2=1-sinθ1+sinθ
Proving secθ-tanθ2=1-sinθ1+sinθ:
Rewrite the left side using trigonometric identities,
secθ-tanθ2=1cosθ-sinθcosθ2
⇒(1-sinθ)2cos2θ
⇒(1-sinθ)21-sin2θ
By perfect square formula,
⇒(1-sinθ)2(1-sinθ)(1+sinθ)
⇒1-sinθ1+sinθ
Hence, it is proved secθ-tanθ2=1-sinθ1+sinθ,.