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.

(* Specifications: *)
Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3.
Lemma sin_PI3 : sin (PI / 3) = sqrt 3 / 2.
Lemma cos_PI3 : cos (PI / 3) = 1 / 2.
Lemma tan_PI3 : tan (PI / 3) = sqrt 3.
Lemma sin_2PI3 : sin (2 * (PI / 3)) = sqrt 3 / 2.
Lemma cos_2PI3 : cos (2 * (PI / 3)) = -1 / 2.
Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3.
Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2.
Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2.
Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)).
Lemma Rgt_3PI2_0 : 0 < 3 * (PI / 2).
Lemma Rgt_2PI_0 : 0 < 2 * PI.
Lemma Rlt_PI_3PI2 : PI < 3 * (PI / 2).
Lemma Rlt_3PI2_2PI : 3 * (PI / 2) < 2 * PI.
Definition plat : R := 180.