apply Rsqr_inj...
- apply cos_ge_0...
+ left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 4) _PI2_RLT_0 PI4_RGT_0)...
+ left; apply PI4_RLT_PI2...
- left; apply (Rmult_lt_0_compat 1 (/ sqrt 2))...
+ prove_sup...
+ apply Rinv_0_lt_compat; apply Rlt_sqrt2_0...
- rewrite Rsqr_div'.
rewrite Rsqr_1; rewrite Rsqr_sqrt...
+ unfold Rsqr; pattern (cos (PI / 4)) at 1;
rewrite <- sin_cos_PI4;
replace (sin (PI / 4) * cos (PI / 4)) with
(1 / 2 * (2 * sin (PI / 4) * cos (PI / 4))) by field.
rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2) by field.
rewrite sin_PI2...
field.
+ left; prove_sup...
Qed.

Lemma cos_PI6 : cos (PI / 6) = sqrt 3 / 2.
Proof with trivial.