Question

(sin70° +cos40°)/(cos70°+sin40°)

Answer 1

Answer

Sin70+cos(90-50)/cos70+sin(90-50)

Sin70+sin50/cos70+cos50

Sin(60+10)+sin(60-10)/cos(60+10)+cos(60-10)

2sin60×cos10/2cos60×cos10

√3/2/1/2=√3

Answer 2

The simplified expression is:

(sin 70° + cos 40°) / (cos 70° + sin 40°) = (sin 55° + cos 40°) / (1 + sin 40°)

(sin70° +cos40°)/(cos70°+sin40°)

We can start by using the following trigonometric identities:

sin (a + b) = sin a cos b + cos a sin b

cos (a + b) = cos a cos b - sin a sin b

Let's rewrite the numerator and denominator using these identities:

sin 70° + cos 40° = sin (70° - 50°) + cos 40°

= sin 70° cos 50° - cos 70° sin 50° + cos 40°

cos 70° + sin 40° = cos (70° - 30°) + sin 40°

= cos 70° cos 30° + sin 70° sin 30° + sin 40°

Now, let's substitute these expressions into the original fraction:

(sin 70° cos 50° - cos 70° sin 50° + cos 40°) / (cos 70° cos 30° + sin 70° sin 30° + sin 40°)

Let's simplify each term:

sin 70° cos 50° = sin (70° + 40°) / 2 = sin 55°

cos 70° sin 50° = cos (50° - 20°) / 2 = cos 35°

cos 70° cos 30° = cos (70° - 60°) / 2 = cos 5°

sin 70° sin 30° = sin (70° - 60°) / 2 = sin 5°

Substituting these values, we get:

(sin 55° + cos 40°) / (cos 5° + sin 5° + sin 40°)

Let's simplify the denominator:

cos 5° + sin 5° = sin (90° - 5°) + sin 5° = sin 90° = 1

Substituting this value, we get:

(sin 55° + cos 40°) / (1 + sin 40°)

Finally,

we can't simplify this any further.

Therefore, the simplified expression is:

(sin 70° + cos 40°) / (cos 70° + sin 40°) = (sin 55° + cos 40°) / (1 + sin 40°)

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