apply Rsqr_inj...
- apply cos_ge_0...
+ left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0)...
+ left; apply PI6_RLT_PI2...
- left; apply (Rmult_lt_0_compat (sqrt 3) (/ 2))...
+ apply Rlt_sqrt3_0...
+ apply Rinv_0_lt_compat; prove_sup0...
- rewrite Rsqr_div'.
rewrite cos2; unfold Rsqr; rewrite sin_PI6; rewrite sqrt_def...
+ field.
+ left ; prove_sup0.
Qed.

Lemma tan_PI3 : tan (PI / 3) = sqrt 3.