Question

The value of expression 1−4sin10∘sin70∘2sin10∘

Answer 1

Given expression is 1−4sin10∘sin70∘2sin10∘1−4sin⁡10∘sin⁡70∘2sin⁡10∘

⇒1−2[cos60∘−cos80∘]2sin10∘⇒1−2[cos⁡60∘−cos⁡80∘]2sin⁡10∘

⇒1−2[12−cos80∘]2sin10∘⇒1−2[12−cos⁡80∘]2sin⁡10∘

⇒2cos80∘2sin10∘⇒2cos⁡80∘2sin⁡10∘

⇒cos(90∘−10∘)sin10∘⇒cos⁡(90∘−10∘)sin⁡10∘

⇒sin10∘sin10∘⇒sin⁡10∘sin⁡10∘

⇒1

Answer 2

Answer:

The value of expression is 4.54.

Step-by-step explanation:

Given : Expression - $1-4(sin10^{\circ})(sin70^{\circ})(2sin10^{\circ})$

To find : The value of the expression

Solution :

Step 1 - Write the expression

$1-4(sin10^{\circ})(sin70^{\circ})(2sin10^{\circ})$

Step 2- Take $2sin10^{\circ}$ common

$1-2.2(sin10^{\circ})[2(sin70^{\circ})(sin10^{\circ})]$

Step 3- Applying property of trigonometry

$2sinAsinB=cos(A-B)-cos(A+B)$

$1-2.2(sin10^{\circ})[cos(70-10)-cos(70+10)]$

$1-2.2(sin10^{\circ})[cos(60)-cos(80)]$

Step 4- Put value of $cos60^\circ=\frac{1}{2}$

$1-4(sin10^{\circ})[\frac{1}{2}-cos(80)]$

$1-2[(sin10^{\circ})-2cos(80^\circ)(sin10^{\circ})$

Step 5- Applying property of trigonometry

$2cosAsinB=sin(A+B)+sin(A-B)$

$1-2[(sin10^{\circ})-(sin(80+10)+sin(80-10))$

$1-2[(sin10^{\circ})-(sin(90)+sin(70))$

Put value of $sin90^\circ=1$

$1-2[(sin10^{\circ})-1-sin(70))$

$1-2(sin10^{\circ}+2+2sin(70)$

$3-2sin10^{\circ}+2sin(70)$

Value of $sin10^\circ=0.17$ and $sin70^\circ=0.94$

$3-2(0.17)+2(0.94)$

$3-0.34+1.88$

$4.54$

The value of expression is 4.54.