MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

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) $t a n 13 θ - t a n 9 θ - t a n 4 θ = t a n 13 θ t a n 9 θ t a n 4 θ$

Open in App

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.

Suggest Corrections

thumbs-up

4

Similar questions

Q. Prove that:
(i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
(ii)

\tan \frac{π}{12} + \tan \frac{π}{6} + \tan \frac{π}{12} \tan \frac{π}{6} = 1

(iii) tan 36° + tan 9° + tan 36° tan 9° = 1
(iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Value of

tan9°+ tan36° + tan9°tan36°

Q. (sin²25° + sin²65°) +

\sqrt{3}

(tan 5° tan 15° tan 30° tan 75° tan 85°)

Q. If the roots of quadratic equation

x^{2} + p x + q = 0

t a n 30^{\circ}

t a n 15^{\circ}

, respectively then the value of

2 + q - p

Q. If the roots of the quadratic equation

x^{2} + p x + q = 0

tan 30^{\circ}

tan 15^{\circ}

respectively . then the value of

2 + q - p

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Trigonometric Ratios for Sum of Two Angles

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon