Question

find the value of: 1/2sin10°-2sin70°​

Answer 1

Answer:

-sec40°/2

OR

1

Step-by-step explanation:

-1/2cos40° if question is 1/(2sin10°- 2sin70°)

and 1 if question is (1/2sin10°) - 2sin70°

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Answer 2

The given trigonometric expression evaluates to 1.

We need to evaluate the value of the given trigonometric expression :

                $(1/2sin10) - 2sin70$

Step 1 : Rationalize by taking LCM in the denominator

We multiply by the term (2sin10°) to both the numerator and denominator of 2sin70°. Thus, the expression becomes :

                   $= (1/2sin10) - 2sin70\\= (1/2sin10) - (2sin70*2sin10)/(2sin10) \\= (1- 4sin70sin10)/(2sin10)$

Step 2 : Using the cos Trigonometric Identity

The identity for cos in terms of the product of sin is :

              $cosA - cosB = 2sin(\frac{A+B}{2} )sin(\frac{B-A}{2} )$

We find two angles An and B such that :

1. (A+B)/2 =  70°
2. (B-A)/2 = 10°

Solving the two equations, we get A = 60° and B = 80°

Thus, applying the identity here :

               $= (1- 2(2sin70sin10))/(2sin10)$

               $= (1- 2(2sin(\frac{140}{2} )sin(\frac{20}{2} ))/(2sin10)$

               $= (1- 2(2sin(\frac{60+80}{2}) sin(\frac{80-60}{2} ))/(2sin10)$

               $= (1 - 2(cos60 - cos80))/2sin10$

Step 3 : Applying trigonometric values

We replace the cos(60°) with its value of 1/2.

                 $= (1 - 2(1/2 - cos80))/2sin10$

Step 4 : Converting cos to sin ratio

The relation between sin and cos is: cosX = sin(90-X)

                  $= (1 - 2(1/2 - sin10))/2sin10$

                  $= (1 - 1 +2sin10))/2sin10$

                  $= ( 2sin10))/2sin10$

                  $=1$

Hence, the given expression evaluates to 1.

To learn more about Trigonometric Identities, visit

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