Question

find the value of sin^2 82 1/2 - sin^2 22 1/2​

Answer 1

Sin ²(82•5) - sin ² (22.5)

sin ²A-sin²b=sin (a+b) sin

(a - b

=sin (82•5122•5) sin +82•5-•25

sin (105) sin

Answer 2

The value of sin² 82 1/2 - sin² 22 1/2​ is equal to  [ (3 +√3) /4√2 ].

Given:

sin² 82 1/2 - sin² 22 1/2​

To find:

The value of sin² 82 1/2 - sin² 22 1/2​

Solution:

Formula used:

From the trigonometric identities

sin²(A) - sin²(B) = [ sin(A + B) × sin(A - B) ]

Here 82 1/2 = 82.5 and 22 1/2 = 22.5°

Let A = 82.5° and B = 22.5°, then:

sin²(82.5°) - sin² (22.5°)

=  [ sin(82.5° + 22.5°) × sin(82.5° - 22.5°) ]

=  [ sin(105°) × sin(60°) ]

We know that

sin(105°) = sin(90 + 15°) = cos(15°)  

sin(60°) = √3/2,  

=> [ sin(105°) × sin(60°) ]  = [ cos(15°) × √3/2 ]

The value of cos(15°) can be find as follows

=> cos (15°) = cos (45° - 30°) = cos (45°)cos(30°) + sin(45°)sin(30°)

= (1/√2 × √3/2) + (1/√2 × 1/2)

= (√3 + 1) / 2√2

Hence, cos (15°)  = (√3 + 1) /2√2

Substitute this value into  [ sin(75°) × √3/2 ]

=  [ (√3 + 1) /2√2 × √3/2 ]

=  [ (3 +√3) /4√2 ]

Therefore,

The value of sin² 82 1/2 - sin² 22 1/2​ is equal to  [ (3 +√3) /4√2 ].

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