(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.
Proof with trivial.
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...
apply sqrt2_neq_0...
Qed.
Notation cos3PI4 := cos_3PI4 (compat "8.10").
Notation sin3PI4 := sin_3PI4 (compat "8.10").
Proof with trivial.
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_PI6 : tan (PI / 6) = 1 / sqrt 3.