Example.1) without using trigonometric tables, evaluate –
sin 35⁰
(i) -------------
cos 55⁰
sin 35⁰ sin (90⁰ - 55⁰) cos 55⁰
Ans.) ------------ = ------------------ = ------------ = 1
cos 55⁰ cos 55⁰ cos 55⁰
sec 41⁰
(ii) -------------
cosec 49⁰
sec 41⁰ sec (90⁰ - 49) cosec 49⁰
------------- = ----------------- = --------------- = 1
cosec 49⁰ cosec 49⁰ cosec 49⁰
cos 35⁰ sin 12 cos 18⁰
Example.2) Find the value of ----------- + ----------- - ----------
sin 55⁰ cos 78⁰ sin 72⁰
cos 35⁰ sin 12 cos 18⁰
Ans.) ------------- + ------------ - -------------
sin 55⁰ cos 78⁰ sin 72⁰
cos (90⁰ - 55⁰) sin (90⁰ - 78⁰) cos (90⁰ - 72⁰)
= ------------------ + ------------------ - ------------------
sin 55⁰ cos 78⁰ sin 72⁰
sin 55⁰ cos 78⁰ sin 72⁰
= -------------- + ------------ - -------------
sin 55⁰ cos 78⁰ sin 72⁰
= 1 + 1 – 1 = 1 (Ans.)
Example.3) without using trigonometric tables, evaluate –
cosec² 57 - tan² 33⁰
Ans.) cosec² 57 - tan² 33⁰
= cosec² 57⁰ - {tan (90⁰ - 57⁰)}² [tan (90⁰ - A) = cot A]
= cosec² 57⁰ - cot² 57⁰
= 1 [cosec² A - cot² A = 1] (Ans.)
Example.4) without using trigonometric tables, prove that –
sec 64⁰ sin 26⁰ + cosec 64⁰ cos 26⁰ = 2
Ans.) sec 64⁰ sin 26⁰ + cosec 64⁰ cos 26⁰
= sec (90⁰ - 26⁰) sin 26⁰ + cosec (90⁰ - 26⁰) cos 26
= cosec 26⁰ sin 26⁰ + sec 26⁰ cos 26⁰ [sec (90⁰ - A) = cosec A, cosec (90⁰ - A) = sec A]
1 1
= (---------- X sin 26⁰) + (---------- X cos 26⁰)
sin 26⁰ cos 26⁰
= (1 + 1) = 1 (Ans.)
Example.5) prove that, cos (60⁰ + A) – sin (30⁰ - A) = 0, where 0⁰ < A < 30⁰
Ans.) we have, LHS = cos (60⁰ + A) – sin (30⁰ - A)
= sin {90⁰ - (60⁰ + A)} – sin (30⁰ - A)
= sin (30⁰ - A) – sin (30⁰ - A)
= 0 (Ans.)
sin A cos A
Example.6) Prove that, ------------ + ------------ = sec A cosec A
sin (90⁰- A) cos (90⁰- A)
Ans.) we have,
sin A cos A
LHS = ------------- + ---------------
sin (90⁰ - A) cos (90⁰ - A)
sin A cos A
= --------- + --------- [sin (90⁰- A) = cos A and cos (90⁰- A)]
cos A sin A
sin² A + cos² A
= -------------------
sin A cos A
1
= ----------------
sin A cos A
1 1
= --------- X ----------
sin A cos A
= sec A cosec A = RHS (Proved)
Example.7) In a △ABC, prove that -
A + B C
sin (---------) = cos ------
2 2
Ans.) As we have A + B + C = 180⁰
So, A + B = (180⁰ - C)
A + B (180⁰ - C)
Or, ----------- = ------------
2 2
A + B C
Or, ---------- = (90⁰ - ------)
2 2
A + B C
Or, sin (---------) = sin (90⁰ - -------)
2 2
A + B C
Or, sin (----------) = cos ------ (Proved)
2 2