Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables: $\;\;\;$ $\dfrac{\sin 110^\circ \sin 250^\circ + \cos 540^\circ \cos 290^\circ \cos 430^\circ}{\cos^2 1260^\circ}$

$\sin 110^\circ = \sin \left(90^\circ + 20^\circ\right) = \cos 20^\circ$

$\sin 250^\circ = \sin \left(270^\circ - 20^\circ\right) = - \cos 20^\circ$

$\cos 540^\circ = \cos \left(2 \times 180^\circ\right) = -1$

$\cos 290^\circ = \cos \left(270^\circ + 20^\circ\right) = \sin 20^\circ$

$\cos 430^\circ = \cos \left(450^\circ - 20^\circ\right) = \sin 20^\circ$

$\cos 1260^\circ = \cos \left(8 \pi - 180^\circ\right) = \cos 180^\circ = -1$

The given expression is: $\;\;$ $\dfrac{\sin 110^\circ \sin 250^\circ + \cos 540^\circ \cos 290^\circ \cos 430^\circ}{\cos^2 1260^\circ}$

$= \dfrac{\cos 20^\circ \times \left(- \cos 20^\circ\right) + \left(-1\right) \times \sin 20^\circ \times \sin 20^\circ}{\left(-1\right)^2}$

$= \dfrac{- \cos^2 20^\circ - \sin^2 20^\circ}{1}$

$= - \left(\cos^2 20^\circ + \sin^2 20^\circ\right) = -1$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables: $\;\;\;$ $\tan 9^\circ - \tan 63^\circ + \tan 81^\circ - \tan 27^\circ$

The given expression is: $\;\;$ $\tan 9^\circ - \tan 63^\circ + \tan 81^\circ - \tan 27^\circ$

$\tan 9^\circ - \tan \left(90^\circ - 27^\circ\right) + \tan \left(90^\circ - 9^\circ\right) - \tan 27^\circ$

$= \dfrac{\sin 9^\circ}{\cos 9^\circ} - \dfrac{\cos 27^\circ}{\sin 27^\circ} + \dfrac{\cos 9^\circ}{\sin 9^\circ} - \dfrac{\sin 27^\circ}{\cos 27^\circ}$

$= \dfrac{\sin^2 9^\circ + \cos^2 9^\circ}{\sin 9^\circ \cos 9^\circ} - \dfrac{\cos^2 27^\circ + \sin^2 27^\circ}{\sin 27^\circ \cos 27^\circ}$

$= \dfrac{1}{\sin 9^\circ \cos 9^\circ} - \dfrac{1}{\sin 27^\circ \cos 27^\circ}$

$= \dfrac{2}{2 \sin 9^\circ \cos 9^\circ} - \dfrac{2}{2 \sin 27^\circ \cos 27^\circ}$

$= \dfrac{2}{\sin \left(2 \times 9^\circ\right)} - \dfrac{2}{\sin \left(2 \times 27^\circ\right)}$

$= 2 \left(\dfrac{1}{\sin 18^\circ} - \dfrac{1}{\sin 54^\circ}\right)$

$= 2 \left(\dfrac{\sin 54^\circ - \sin 18^\circ}{\sin 54^\circ \sin 18^\circ}\right)$

$= \dfrac{2 \times 2 \sin \left(\dfrac{54^\circ - 18^\circ}{2}\right) \cos \left(\dfrac{54^\circ + 18^\circ}{2}\right)}{\sin 54^\circ \sin 18^\circ}$

$= \dfrac{4 \sin 18^\circ \cos 36^\circ}{\sin 54^\circ \sin 18^\circ}$

$= \dfrac{4 \cos 36^\circ}{\sin 54^\circ}$

$= \dfrac{4 \cos \left(90^\circ - 54^\circ\right)}{\sin 54^\circ}$

$= \dfrac{4 \sin 54^\circ}{\sin 54^\circ} = 4$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables:
$\cos 10^\circ \cos 50^\circ \cos 70^\circ$

The given expression is: $\;\;$ $\cos 10^\circ \cos 50^\circ \cos 70^\circ$

$= \dfrac{1}{2} \times \left[2 \cos 10^\circ \cos 50^\circ\right] \cos 70^\circ$

$= \dfrac{1}{2} \left[\cos \left(10^\circ - 50^\circ\right) + \cos \left(10^\circ + 50^\circ\right)\right] \cos 70^\circ$

$= \dfrac{1}{2} \left[\cos \left(-40^\circ\right) + \cos 60^\circ\right] \cos 70^\circ$

$= \dfrac{1}{2} \cos 70^\circ \cos 40^\circ + \dfrac{1}{2} \cos 60^\circ \cos 70^\circ$

$= \dfrac{1}{4} \left[2 \cos 70^\circ \cos 40^\circ\right] + \dfrac{1}{2} \times \dfrac{1}{2} \times \cos 70^\circ$

$= \dfrac{1}{4} \left[\cos \left(70^\circ - 40^\circ\right) + \cos \left(70^\circ + 40^\circ\right)\right] + \dfrac{1}{4} \times \cos 70^\circ$

$= \dfrac{1}{4} \left[\cos 30^\circ + \cos 110^\circ\right] + \dfrac{1}{4} \cos 70^\circ$

$= \dfrac{1}{4} \times \dfrac{\sqrt{3}}{2} + \dfrac{1}{4} \times \cos 110^\circ + \dfrac{1}{4} \cos 70^\circ$

$= \dfrac{\sqrt{3}}{8} + \dfrac{1}{4} \left[\cos 110^\circ + \cos 70^\circ\right]$

$= \dfrac{\sqrt{3}}{8} + \dfrac{1}{4} \left[\cos \left(180^\circ - 70^\circ\right) + \cos 70^\circ\right]$

$= \dfrac{\sqrt{3}}{8} + \dfrac{1}{4} \left[\cos \left(- 70^\circ\right) + \cos 70^\circ\right]$

$= \dfrac{\sqrt{3}}{8} + \dfrac{1}{4} \left[- \cos 70^\circ + \cos 70 ^\circ\right]$

$= \dfrac{\sqrt{3}}{8}$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables:
$\tan 20^\circ \tan 40^\circ \tan 80^\circ$

The given expression is: $\;\;$ $\tan 20^\circ \tan 40^\circ \tan 80^\circ$

$= \dfrac{\sin 20^\circ \sin 40^\circ \sin 80^\circ}{\cos 20^\circ \cos 40^\circ \cos 80^\circ}$

$= \dfrac{\sin 20^\circ \times \dfrac{1}{2} \left[2 \sin 80^\circ \sin 40^\circ\right]}{\cos 20^\circ \times \dfrac{1}{2} \left[2 \cos 80^\circ \cos 40^\circ\right]}$

$= \dfrac{\sin 20^\circ \times \dfrac{1}{2} \left[\cos \left(80^\circ - 40^\circ\right) - \cos \left(80^\circ + 40^\circ\right)\right]}{\cos 20^\circ \times \dfrac{1}{2} \left[\cos \left(80^\circ - 40^\circ\right) + \cos \left(80^\circ + 40^\circ\right)\right]}$

$= \dfrac{\sin 20^\circ \left[\cos 40^\circ - \cos 120^\circ\right]}{\cos 20^\circ \left[\cos 40^\circ + \cos 120^\circ\right]}$

$= \dfrac{\sin 20^\circ \left[\cos 40^\circ - \left(\dfrac{-1}{2}\right)\right]}{\cos 20^\circ \left[\cos 40^\circ + \left(\dfrac{-1}{2}\right)\right]}$

$= \dfrac{2 \sin 20^\circ \cos 40^\circ + \sin 20^\circ}{2 \cos 20^\circ \cos 40^\circ - \cos 20^\circ}$

$= \dfrac{\sin \left(40^\circ + 20^\circ\right) + \sin \left(20^\circ - 40^\circ\right) + \sin 20^\circ}{\cos \left(40^\circ - 20^\circ\right) + \cos \left(40^\circ + 20^\circ\right) - \cos 20^\circ}$

$= \dfrac{\sin 60^\circ + \sin \left(-20^\circ\right) + \sin 20^\circ}{\sin 20^\circ + \cos 60^\circ - \cos 20^\circ}$

$= \dfrac{\sin 60^\circ - \sin 20^\circ + \sin 20^\circ}{\cos 60^\circ}$

$= \dfrac{\sin 60^\circ}{\cos 60^\circ}$

$= \tan 60^\circ = \sqrt{3}$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables:
$128 \sin^2 20^\circ \sin^2 40^\circ \sin^2 60^\circ \sin^2 80^\circ$

The given expression is: $\;\;$ $128 \sin^2 20^\circ \sin^2 40^\circ \sin^2 60^\circ \sin^2 80^\circ$ $\;\;\; \cdots \; (1)$

Consider the expression: $\;\;$ $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ$

$= \sin 20^\circ \times \left(\sin 80^\circ \sin 40^\circ\right) \times \dfrac{\sqrt{3}}{2}$

$= \dfrac{\sqrt{3}}{2} \times \sin 20^\circ \times \dfrac{1}{2} \left[\cos \left(80^\circ - 40^\circ\right) - \cos \left(80^\circ + 40^\circ\right)\right]$

$= \dfrac{\sqrt{3}}{4} \times \sin 20^\circ \left[\cos 40^\circ - \cos 120^\circ\right]$

$= \dfrac{\sqrt{3}}{4} \times \sin 20^\circ \left[\cos 40^\circ - \left(\dfrac{-1}{2}\right)\right]$

$= \dfrac{\sqrt{3}}{8} \left[2 \sin 20^\circ \cos 40^\circ + \sin 20^\circ\right]$

$= \dfrac{\sqrt{3}}{8} \left[\sin \left(20^\circ + 40^\circ\right) + \sin \left(20^\circ - 40^\circ\right) + \sin 20^\circ\right]$

$= \dfrac{\sqrt{3}}{8} \left[\sin 60^\circ + \sin \left(-20^\circ\right) + \sin 20^\circ\right]$

$= \dfrac{\sqrt{3}}{8} \left[\dfrac{\sqrt{3}}{2} - \sin 20^\circ + \sin 20^\circ\right]$

$\therefore$ $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \dfrac{3}{16}$ $\;\;\; \cdots \; (2)$

In view of expression $(2)$, expression $(1)$ becomes

$128 \times \left(\dfrac{3}{16}\right)^2 = \dfrac{9}{2}$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables:
$\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ}$

The given expression is: $\;\;$ $\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ}$ $\;\;\; \cdots \; (1)$

Denominator of the given expression is

$\sin 20^\circ + \sin 50^\circ + \sin 110^\circ$

$= \sin \left(2 \times 10^\circ\right) + \left(\sin 50^\circ + \sin 110^\circ\right)$

$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin \left(\dfrac{110^\circ + 50^\circ}{2}\right) \cos \left(\dfrac{110^\circ - 50^\circ}{2}\right)$

$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin 80^\circ \cos 30^\circ$

$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin \left(90^\circ - 10^\circ\right) \cos \left(90^\circ - 60^\circ\right)$

$= 2 \sin 10^\circ \cos 10^\circ + 2 \cos 10^\circ \sin 60^\circ$

$= 2 \cos 10^\circ \left(\sin 60^\circ + \sin 10^\circ\right)$

$= 2 \cos \left(90^\circ - 80^\circ\right) \times 2 \sin \left(\dfrac{60^\circ + 10^\circ}{2}\right) \cos \left(\dfrac{60^\circ - 10^\circ}{2}\right)$

$= 4 \sin 80^\circ \sin 35^\circ \cos 25^\circ$

$= 4 \sin 80^\circ \sin 35^\circ \cos \left(90^\circ - 65^\circ\right)$

$= 4 \sin 80^\circ \sin 35^\circ \sin 65^\circ$ $\;\;\; \cdots \; (2)$

In view of $(2)$, the given expression $(1)$ becomes

$\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{4 \sin 80^\circ \sin 35^\circ \sin 65^\circ}$

$= \dfrac{96}{4} = 24$

Trigonometry - Simplification of Trigonometric Expressions

Calculate without using tables:
$4 \left(\cos 24^\circ + \cos 48^\circ - \cos 84^\circ - \cos 12^\circ\right)$

$4 \left(\cos 24^\circ + \cos 48^\circ - \cos 84^\circ - \cos 12^\circ\right)$

$= 4 \left[\left(\cos 24^\circ - \cos 84^\circ\right) + \left(\cos 48^\circ - \cos 12^\circ\right)\right]$

$= 4 \left[-2 \sin \left(\dfrac{24^\circ - 84^\circ}{2}\right) \sin \left(\dfrac{24^\circ + 84^\circ}{2}\right) - 2 \sin \left(\dfrac{48^\circ - 12^\circ}{2}\right) \sin \left(\dfrac{48^\circ + 12^\circ}{2}\right)\right]$

$= 4 \times \left(-2\right) \left[\sin \left(-30^\circ\right) \sin 54^\circ + \sin 18^\circ \sin 30^\circ\right]$

$= -8 \left[\dfrac{-1}{2} \times \sin 54^\circ + \dfrac{1}{2} \times \sin 18^\circ\right]$

$= -8 \times \left(\dfrac{-1}{2}\right) \left[\sin 54^\circ - \sin 18^\circ\right]$

$= 4 \left[\sin 54^\circ - \sin 18^\circ\right]$

$= 4 \times 2 \times \left(\dfrac{54^\circ - 18^\circ}{2}\right) \cos \left(\dfrac{54^\circ + 18^\circ}{2}\right)$

$= 4 \times 2 \times \sin 18^\circ \cos 36^\circ$ $\;\;\; \cdots \; (1)$

Now, $\;$ $\sin 36^\circ = \sin \left(2 \times 18^\circ\right) = 2 \sin 18^\circ \cos 18^\circ$

$\implies$ $\sin 18^\circ = \dfrac{\sin 36^\circ}{2 \cos 18^\circ}$ $\;\;\; \cdots \; (2)$

In view of $(2)$, expression $(1)$ becomes

$4 \times 2 \cos 36^\circ \times \dfrac{\sin 36^\circ}{2 \cos 18^\circ}$

$= \dfrac{2 \sin \left(2 \times 36^\circ\right)}{\cos 18^\circ}$

$= \dfrac{2 \sin 72^\circ}{\cos \left(90^\circ - 72^\circ\right)}$

$= \dfrac{2 \sin 72^\circ}{\sin 72^\circ}$

$= 2$