Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\tan \left(\dfrac{\alpha}{2} + \dfrac{\pi}{4}\right) \left(\dfrac{1 - \sin \alpha}{\cos \alpha}\right)$

$\tan \left(\dfrac{\alpha}{2} + \dfrac{\pi}{4}\right) \left(\dfrac{1 - \sin \alpha}{\cos \alpha}\right)$

$= \left(\dfrac{\tan \dfrac{\pi}{4} + \tan \dfrac{\alpha}{2}}{1 - \tan \dfrac{\pi}{4} \times \tan \dfrac{\alpha}{2}}\right) \left(\dfrac{\sin^2 \dfrac{\alpha}{2} + \cos^2 \dfrac{\alpha}{2} - 2 \sin \dfrac{\alpha}{2} \cos \dfrac{\alpha}{2}}{\cos^2 \dfrac{\alpha}{2} - \sin^2 \dfrac{\alpha}{2}}\right)$

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

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

$= \left(\dfrac{\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}}\right) \times \left(\dfrac{\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}}\right)$

$= 1$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\dfrac{2 \cos^2 \alpha - 1}{4 \tan \left(\dfrac{\pi}{4} - \alpha\right) \sin^2 \left(\dfrac{\pi}{4} + \alpha\right)}$

$\dfrac{2 \cos^2 \alpha - 1}{4 \tan \left(\dfrac{\pi}{4} - \alpha\right) \sin^2 \left(\dfrac{\pi}{4} + \alpha\right)}$

$= \dfrac{\cos 2 \alpha}{4 \left[\dfrac{\tan \dfrac{\pi}{4} - \tan \alpha}{1 + \tan \dfrac{\pi}{4} \tan \alpha}\right] \left[\sin \dfrac{\pi}{4} \cos \alpha + \cos \dfrac{\pi}{4} \sin \alpha\right]^2}$

$= \dfrac{\cos 2 \alpha}{4 \left[\dfrac{1 - \tan \alpha}{1 + \tan \alpha}\right] \left[\dfrac{1}{\sqrt{2}} \cos \alpha + \dfrac{1}{\sqrt{2}} \sin \alpha\right]^2}$

$= \dfrac{\cos 2 \alpha}{4 \left[\dfrac{1 - \dfrac{\sin \alpha}{\cos \alpha}}{1 + \dfrac{\sin \alpha}{\cos \alpha}}\right] \left[\dfrac{\cos^2 \alpha}{2} + \dfrac{\sin^2 \alpha}{2} + 2 \times \dfrac{1}{\sqrt{2}} \cos \alpha \times \dfrac{1}{\sqrt{2}} \sin \alpha\right]}$

$= \dfrac{\cos 2 \alpha}{4 \left[\dfrac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha}\right] \left[\dfrac{1}{2} + \sin \alpha \cos \alpha\right]}$

$= \dfrac{\cos 2 \alpha}{4 \times \dfrac{\left(\cos \alpha - \sin \alpha\right) \left(\cos \alpha + \sin \alpha\right)}{\left(\cos \alpha + \sin \alpha\right)^2} \times \left(\dfrac{1 + 2 \sin \alpha \cos \alpha}{2}\right)}$

$= \dfrac{\cos 2 \alpha \times \left(\cos \alpha + \sin \alpha\right)^2 \times 2}{4 \times \left(\cos^2 \alpha - \sin^2 \alpha\right) \times \left(\sin^2 \alpha + \cos^2 \alpha + 2 \sin \alpha \cos \alpha\right)}$

$= \dfrac{\cos 2 \alpha \times \left(\cos \alpha + \sin \alpha\right)^2}{2 \times \cos 2 \alpha \times \left(\cos \alpha + \sin \alpha\right)^2}$

$= \dfrac{1}{2}$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\cos 0 + \cos \dfrac{\pi}{7} + \cos \dfrac{2 \pi}{7} + \cos \dfrac{3 \pi}{7} + \cos \dfrac{4 \pi}{7} + \cos \dfrac{5 \pi}{7} + \cos \dfrac{6 \pi}{7}$

$\cos 0 + \cos \dfrac{\pi}{7} + \cos \dfrac{2 \pi}{7} + \cos \dfrac{3 \pi}{7} + \cos \dfrac{4 \pi}{7} + \cos \dfrac{5 \pi}{7} + \cos \dfrac{6 \pi}{7}$

$= 1 + \left(\cos \dfrac{6 \pi}{7} + \cos \dfrac{\pi}{7}\right) + \left(\cos \dfrac{5 \pi}{7} + \cos \dfrac{2 \pi}{7}\right) + \left(\cos \dfrac{4 \pi}{7} + \cos \dfrac{3 \pi}{7}\right)$

$= 1 + 2 \cos \left(\dfrac{\dfrac{6 \pi}{7} + \dfrac{\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{6 \pi}{7} - \dfrac{\pi}{7}}{2}\right)$
$\hspace{1.5cm} + 2 \cos \left(\dfrac{\dfrac{5 \pi}{7} + \dfrac{2\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{5 \pi}{7} - \dfrac{2\pi}{7}}{2}\right)$
$\hspace{2.5cm} + 2 \cos \left(\dfrac{\dfrac{4\pi}{7} + \dfrac{3\pi}{7}}{2}\right) \cos \left(\dfrac{\dfrac{4\pi}{7} - \dfrac{3\pi}{7}}{2}\right)$

$= 1 + 2 \cos \dfrac{\pi}{2} \cos \dfrac{5 \pi}{14} + 2 \cos \dfrac{\pi}{2} \cos \dfrac{3 \pi}{14} + 2 \cos \dfrac{\pi}{2} \cos \dfrac{\pi}{14}$

$= 1 + 2 \times 0 \times \cos \dfrac{5 \pi}{14} + 2 \times 0 \times \cos \dfrac{3\pi}{14} + 2 \times 0 \times \cos \dfrac{\pi}{14}$

$= 1$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\dfrac{\sin 3 \alpha + \sin 5 \alpha + \sin 7 \alpha}{\cos 3 \alpha + \cos 5 \alpha + \cos 7 \alpha}$

$\dfrac{\sin 3 \alpha + \sin 5 \alpha + \sin 7 \alpha}{\cos 3 \alpha + \cos 5 \alpha + \cos 7 \alpha}$

$= \dfrac{\left(\sin 3 \alpha + \sin 7 \alpha\right) + \sin 5 \alpha}{\left(\cos 3 \alpha + \cos 7 \alpha\right) + \cos 5 \alpha}$

$= \dfrac{2 \sin \left(\dfrac{7 \alpha + 3 \alpha}{2}\right) \cos \left(\dfrac{7 \alpha - 3 \alpha}{2}\right) + \sin 5 \alpha}{2 \cos \left(\dfrac{7 \alpha + 3 \alpha}{2}\right) \cos \left(\dfrac{7 \alpha - 3 \alpha}{2}\right) + \cos 5 \alpha}$

$= \dfrac{2 \sin 5 \alpha \cos 2 \alpha + \sin 5 \alpha}{2 \cos 5 \alpha \cos 2 \alpha + \cos 5 \alpha}$

$= \dfrac{\sin 5 \alpha \left(2 \cos 2 \alpha + 1\right)}{\cos 5 \alpha \left(2 \cos 2 \alpha + 1\right)}$

$= \dfrac{\sin 5 \alpha}{\cos 5 \alpha}$

$= \tan 5 \alpha$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\dfrac{1 - \cos 4 \alpha}{\sec^2 2 \alpha - 1} + \dfrac{1 + \cos 4 \alpha}{\text{cosec}^2 2 \alpha - 1}$

$\dfrac{1 - \cos 4 \alpha}{\sec^2 2 \alpha - 1} + \dfrac{1 + \cos 4 \alpha}{\text{cosec}^2 2 \alpha - 1}$

$= \dfrac{2 \sin^2 2 \alpha}{\dfrac{1}{\cos^2 2 \alpha} - 1} + \dfrac{2 \cos^2 2 \alpha}{\dfrac{1}{\sin^2 2 \alpha} - 1}$

$\left[\because \; 1 - \cos 2 \theta = 2 \sin^2 \theta \implies 1 - \cos 4 \theta = 2 \sin^2 2 \theta\right] \;\;\;$ and

$\left[1 + \cos 2 \theta = 2 \cos^2 \theta \implies 1 + \cos 4 \theta = 2 \cos^2 2 \theta\right]$

$= \dfrac{2 \sin^2 2 \alpha \cos^2 2 \alpha}{1 - \cos^2 2 \alpha} + \dfrac{2 \cos^2 2 \alpha \sin^2 2 \alpha}{1 - \sin^2 2 \alpha}$

$= \dfrac{2 \sin^2 2 \alpha \cos^2 2 \alpha}{\sin^2 2 \alpha} + \dfrac{2 \cos^2 2 \alpha \sin^2 2 \alpha}{\cos^2 2 \alpha}$

$= 2 \sin^2 2 \alpha \cos^2 2 \alpha \left(\dfrac{1}{\sin^2 2 \alpha} + \dfrac{1}{\cos^2 2 \alpha}\right)$

$= 2 \sin^2 2 \alpha \cos^2 2 \alpha \left(\dfrac{\cos^2 2 \alpha + \sin^2 2 \alpha}{\sin^2 2 \alpha \cos^2 2 \alpha}\right)$

$= 2 \times 1 = 2$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $\dfrac{\tan \alpha + \sin \alpha}{2 \cos^2 \left(\dfrac{\alpha}{2}\right)}$

$\dfrac{\tan \alpha + \sin \alpha}{2 \cos^2 \left(\dfrac{\alpha}{2}\right)}$

$= \dfrac{\tan \alpha + \sin \alpha}{1 + \cos \alpha}$

$= \dfrac{\dfrac{\sin \alpha}{\cos \alpha} + \sin \alpha}{1 + \cos \alpha}$

$= \dfrac{\sin \alpha \left(1 + \cos \alpha\right)}{\cos \alpha \left(1 + \cos \alpha\right)}$

$= \dfrac{\sin \alpha}{\cos \alpha}$

$= \tan \alpha$

Trigonometry - Simplification of Trigonometric Expressions

Simplify the expression: $3 \cos^2 x + 4 \sin x \cos x - \sin^2 x - 1$

$3 \cos^2 x + 4 \sin x \cos x - \sin^2 x - 1$

$= 2 \cos^2 x + 2 \times 2 \sin x \cos x - \sin^2 x - \left(1 - \cos^2 x\right)$

$= 2 \cos^2 x + 2 \sin 2 x - \sin^2 x - \sin^2 x$

$= 2 \cos^2 x + 2 \sin 2 x - 2 \sin^2 x$

$= 2 \left(\cos^2 x - \sin^2 x\right) + 2 \sin 2 x$

$= 2 \left(\cos 2 x + \sin 2 x\right)$

$= 2 \sqrt{2} \left(\cos 2 x \times \dfrac{1}{\sqrt{2}} + \sin 2 x \times \dfrac{1}{\sqrt{2}}\right)$

$= 2 \sqrt{2} \left(\sin 2 x \times \cos \dfrac{\pi}{4} + \cos 2 x \times \sin \dfrac{\pi}{4}\right)$

$= 2 \sqrt{2} \sin \left(2x + \dfrac{\pi}{4}\right)$