Question

|sinx • cosx| + $\sqrt{ { \tan(x) }^{2} { \cot(x) }^{2} + 2 \tan(x \cot(x) ) } = \sqrt{3}$
find the number of real solutions of x ​

Answer 1

Answer:

No solution

Step-by-step explanation:

Given:

|sinx • cosx| + $\sqrt{ { \tan(x) }^{2} { \cot(x) }^{2} + 2 \tan(x \cot(x) ) } = \sqrt{3}$

Solve as follows:

Multiply and divide by 2 in first term

$\frac{1}{2} \times | 2\sin(x) \times \cos(x) | + \sqrt{ { \tan }^{2} x + { \cot }^{2} x + 2 } = \sqrt{3}$

Now as Tanx = 1/cotx

As Sin(2x) = 2sin(x)×cos(x)

$\frac{1}{2} | \sin(2x) | + \sqrt{ { \tan}^{2} x + { \cot }^{2} x + 2 \tan(x) \cot(x) } = \sqrt{3} \\ \\ \frac{1}{2} | \sin(2x) | + \sqrt{ {( \tan(x) + \cot(x) ) }^{2} } = \sqrt{3}$

Now, Quantity under square root is always positive so it will opens with +ve value

$\frac{1}{2} | \sin(2x) | + | \tan(x) + \cot(x) | = \sqrt{3} \\ \\ \frac{1}{2} | \sin(2x) | + | \tan(x) + \frac{1}{ \tan(x) ) } | = \sqrt{3} \\ \\ \frac{1}{2} | \sin(2x) | + | \frac{ { \tan }^{2}x + 1 }{ \tan(x) } | = \sqrt{3}$

Now use the formula,

$\sin(2x) = \frac{2 \tan(x) }{1 + { \tan }^{2}x }$

Putting in

$\frac{1}{2} | \frac{2 \tan(x) }{1 + { \tan }^{2} x} | + | \frac{ { \tan }^{2}x + 1 }{ \tan(x) } | = \sqrt{3}$

Now replace $| \dfrac{ { \tan }^{2}x + 1 }{ \tan(x) } |$ by t

Equation becomes

$\frac{1}{2} | \frac{2 \tan(x) }{1 + { \tan }^{2} x} | + | \frac{ { \tan }^{2}x + 1 }{ \tan(x) } | = \sqrt{3} \\ \\ t + \frac{1}{t} = \sqrt{3} \\ \\ \frac{ {t}^{2} + 1 }{t} = \sqrt{3} \\ \\ {t}^{2} + 1 = \sqrt{3} t \\ \\ {t}^{2} - \sqrt{3} t + 1 = 0$

Now here we get quadratic equation in t

To find number of solution first check the existence of solutions

• Check Discriminate of the equation

Using $d = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a}$

Putting in the values

$d = \frac{ + \sqrt{3} \pm \sqrt{3 - 4 \times 1 \times 1} }{2} \\ \\ d = \frac{ \sqrt{3} \pm \sqrt{ - 1} }{2}$

The quantity under squareroot is negative so we get D <0

Hence no real solutions exist.

Answer No solution

$\boxed{\underline{\underline{\bold{\mathsf{Formula\:Used\:here\:are:}}}}}$:—

•Sin(2x) = 2sin(x)×cos(x)

• $\sin(2x) = \frac{2 \tan(x) }{1 + { \tan }^{2}x }$

•$d = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a}$

Answer 2

Hey there

refer to attachment