The number of solution in -0- - - 2- of the equation cos 3 x tan 5 x-sin 7 x is

The correct option is D 6 Given: cos 3x tan 5x=sin7x ⇒ cos (5x 2x)tan 5x=sin(5x+2x) ⇒ tan 5x=sin(5x+2x)/cos(5x 2x) ⇒ tan 5x=sin 5x cos2x+cos5x sin 2x/cos 5x c ...

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Byju's Answer

Standard XII

Mathematics

General Solution of Trigonometric Equation

The number of...

Question

$The number of solution in [0, π / 2] o f t h e e q u a t i o n c o s 3 x t a n 5 x = s i n 7 x i s$

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Solution

The correct option is C
6

$G i v e n : c o s 3 x t a n 5 x = s i n 7 x \Rightarrow c o s (5 x - 2 x) t a n 5 x = s i n (5 x + 2 x) \Rightarrow t a n 5 x = \frac{s i n (5 x + 2 x)}{c o s (5 x - 2 x)} \Rightarrow t a n 5 x = \frac{s i n 5 x c o s 2 x + c o s 5 x s i n 2 x}{c o s 5 x c o s 2 x + s i n 5 x s i n 2 x} \Rightarrow \frac{s i n 5 x}{c o s 5 x} = \frac{s i n 5 x c o s 2 x + c o s 5 x s i n 2 x}{c o s 5 x c o s 2 x + s i n 5 x s i n 2 x} \Rightarrow s i n 5 x c o s 5 x c o s 2 x + s i n^{2} 5 x s i n 2 x = s i n 5 x c o s 5 x c o s 2 x + c o s^{2} 5 x s i n 2 x \Rightarrow s i n^{2} 5 x s i n 2 x = c o s^{2} 5 x s i n 2 x \Rightarrow (s i n^{2} 5 x - c o s^{2} 5 x) s i n 2 x = 0 \Rightarrow (s i n 5 x - c o s 5 x) (s i n 5 x + c o s 5 x) s i n 2 x = 0 \Rightarrow s i n 5 x - c o s 5 x = 0, s i n 5 x + c o s 5 x = 0 o r s i n 2 x = 0 \Rightarrow \frac{s i n 5 x}{c o s 5 x} = 1, \frac{s i n 5 x}{c o s 5 x} = - 1 o r s i n 2 x = 0 N o w, t a n 5 x = 1 \Rightarrow t a n 5 x = t a n \frac{π}{4} \Rightarrow 5 x = n π + \frac{π}{4}, n \in Z \Rightarrow x = \frac{n π}{5} + \frac{π}{20}, n \in Z F o r n = 0, 1 a n d 2, t h e v a l u e o f x a r e \frac{π}{20}, \frac{π}{4} a n d \frac{9 π}{20}, r e s p e c t i v e l y o r t a n 5 x = 1 \Rightarrow t a n 5 x = t a n \frac{3 π}{4} \Rightarrow 5 x = n π + \frac{3 π}{4}, n \in Z \Rightarrow x = \frac{n π}{5} + \frac{3 π}{20}, n \in Z F o r n = 0 a n d 1, t h e v a l u e s o f x a r e \frac{3 π}{20} a n d \frac{7 π}{20}, r e s p e c t i v e l y . A n d, s i n 2 x = 0 \Rightarrow s i n 2 x = s i n 0 \Rightarrow 2 x = n π, n \in Z \Rightarrow x = \frac{n π}{2}, n π Z F o r n = 0 t h e v a l u e o f x i s 0. A l s o, f o r t h e o d d m u l t i p l e o f \frac{π}{2}, t a n x i s n o t d e f i n e d . H e n c e, t h e r e a r e s i x s o l u t i o n s .$

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The number of solution in[0,π/2]of the equation cos 3x tan 5x=sin7x is

$The number of solution in [0, π / 2] o f t h e e q u a t i o n c o s 3 x t a n 5 x = s i n 7 x i s$