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Trigonometric Ratios for Sum of Two Angles

Prove that:i ...

Question

Prove that:
(i) $t a n 8 θ - t a n 6 θ - t a n 2 θ = t a n 8 θ t a n 6 θ t a n 2 θ$
(ii) $t a n 15^{\circ} + t a n 30^{\circ} + t a n 15^{\circ} t a n 30^{\circ} = 1$
(iii) $t a n 36^{\circ} + t a n 9^{\circ} + t a n 36^{\circ} t a n 9^{\circ} = 1$
(iv) $t a n 13 θ - t a n 9 θ - t a n 4 θ = t a n 13 θ t a n 9 θ t a n 4 θ$

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Solution

(i) We have,
$8 θ = 6 θ + 2 θ$
$\Rightarrow t a n 8 θ = t a n (6 θ + 2 θ)$
$\Rightarrow t a n 8 θ = \frac{t a n 6 θ + t a n 2 θ}{1 - t a n 6 θ t a n 2 θ}$
$\Rightarrow t a n 8 θ (1 - t a n 6 θ t a n 2 θ) = t a n 6 θ + t a n 2 θ$
$\Rightarrow t a n 8 θ - t a n 8 θ t a n 6 θ t a n 2 θ = t a n 6 θ + t a n 2 θ$
$\Rightarrow t a n 8 θ - t a n 6 θ - t a n 2 θ = t a n 8 θ t a n 6 θ t a n 2 θ$
Hence proved.
(ii) We have, $45^{\circ} = t a n (30^{\circ} + 15^{\circ})$
$\Rightarrow 1 = \frac{t a n 30^{\circ} + t a n 15^{\circ}}{1 - t a n 30^{\circ} t a n 15^{\circ}}$
$\Rightarrow 1 - t a n 30^{\circ} t a n 15^{\circ} = t a n 15^{\circ} + t a n 30^{\circ}$
$\Rightarrow 1 = t a n 15^{\circ} + t a n 30^{\circ} + t a n 30^{\circ} t a n 15^{\circ}$
$\Rightarrow t a n 15^{\circ} + t a n 30^{\circ} + t a n 15^{\circ} t a n 30^{\circ} = 1$
Hence proved.
(iii) We have,
$45^{\circ} = 9^{\circ} + 36$
$\Rightarrow t a n 45^{\circ} = t a n (9^{\circ} + 36^{\circ})$
$\Rightarrow 1 = \frac{t a n 9^{\circ} + t a n 36^{\circ}}{1 - t a n 9^{\circ} t a n 36^{\circ}}$
$\Rightarrow 1 - t a n 9^{\circ} t a n 36^{\circ} = t a n 9^{\circ} + t a n 36^{\circ}$
$\Rightarrow 1 = t a n 9^{\circ} + t a n 36^{\circ} + t a n 9^{\circ} t a n 36^{\circ}$
$\Rightarrow t a n 9^{\circ} + t a n 36^{\circ} + t a n 9^{\circ} t a n 36^{\circ} = 1$
Hence proved.
(iv)We have,
$13 θ = 9 θ + 4 θ$
$\Rightarrow t a n 13 θ = t a n (9 θ + 4 θ)$
$\Rightarrow 13 θ = \frac{t a n 9 θ + t a n 4 θ}{1 - t a n 9 θ t a n 4 θ}$
$\Rightarrow t a n 13 θ (1 - t a n 9 θ t a n 4 θ) = t a n 9 θ + t a n 4 θ$
$\Rightarrow t a n 13 θ - t a n 13 θ t a n 9 θ t a n 4 θ = t a n 9 θ + t a n 4 θ$
$\Rightarrow t a n 13 θ - t a n 9 θ - t a n 4 θ = t a n 13 θ t a n 9 θ t a n 4 θ$
Hence proved.

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