If x is an acute angle such that both tan kx and kx is defined for k-0-20- then which of the following is-are correct -

Tan15x=tan(11x+4x) ⇒tan15x=tan11x+tan4x1 tan11xtan4x ⇒tan15x tan15xtan11xtan4x=tan11x+tan4x ⇒tan15x tan11x tan4x=tan15xtan11xtan4x 6x=(2x+4x) ⇒6x=2x 4x 12x+ 4x ...

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Byju's Answer

Standard XIII

Mathematics

Trigonometric Ratios for Sum of Two Angles

If x is an ac...

Question

If $x$ is an acute angle such that both $tan (k x)$ and $cot (k x)$ is defined for $k \in (0, 20),$ then which of the following is/are correct ?

tan 15 x + tan 11 x + tan 4 x = tan 15 x tan 11 x tan 4 x

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tan 15 x - tan 11 x - tan 4 x = tan 15 x tan 11 x tan 4 x

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cot 2 x cot 4 x - cot 4 x cot 6 x - cot 2 x cot 6 x = 1

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cot 4 x cot 6 x - cot 2 x cot 4 x - cot 2 x cot 6 x = 1

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Solution

The correct option is C $cot 2 x cot 4 x - cot 4 x cot 6 x - cot 2 x cot 6 x = 1$
$tan 15 x = tan (11 x + 4 x)$
$\Rightarrow tan 15 x = \frac{tan 11 x + tan 4 x}{1 - tan 11 x tan 4 x}$
$\Rightarrow tan 15 x - tan 15 x tan 11 x tan 4 x = tan 11 x + tan 4 x$
$\Rightarrow tan 15 x - tan 11 x - tan 4 x = tan 15 x tan 11 x tan 4 x$

$cot 6 x = cot (2 x + 4 x)$
$\Rightarrow cot 6 x = \frac{cot 2 x cot 4 x - 1}{cot 2 x + cot 4 x}$
$\Rightarrow cot 6 x cot 2 x + cot 6 x cot 4 x = cot 2 x cot 4 x - 1$
$\Rightarrow cot 2 x cot 4 x - cot 4 x cot 6 x - cot 2 x cot 6 x = 1$

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Similar questions

Q. If

x

is an acute angle such that both

tan (k x)

and

cot (k x)

is defined for

k \in (0, 20),

then which of the following is/are correct ?

Question 5 (x)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\frac{1 + t a n^{2} A}{1 + c o t^{2} A}) = (\frac{1 - t a n A}{1 - c o t A})^{2} = t a n^{2} A$