Advertisements
Advertisements
Question
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Advertisements
Solution
Step 1: Rewrite the tangent function
We know that:
tan θ = `sin θ/cos θ`
Thus, we can rewrite the left-hand side (LHS) as:
tan 20° tan 30° tan 40° tan 80° = `sin 20^@/cos 20^@ · sin 30^@/cos 30^@ · sin 40^@/cos^@ · sin 80^@/cos 80^@`
This can be simplified to:
`sin 20^@ sin 30^@ sin 40^@ sin 80^@/cos 20^@ cos 30^@ cos 40^@ cos 80^@`
Step 2: Use known values
We know that:
`sin 30^@ = 1/2 and cos 30^@ = sqrt3/2`
Substituting these values into the equation gives us:
= `(sin 20^@ · 1/2 · sin 40^@ · sin 80^@)/(cos 20^@ · sqrt3/2 · cos 40^@ · cos 80^@)`
This simplifies to:
= `sin 20^@ sin 40^@ sin 80^@/cos 20^@ cos 40^@ cos 80^@ · 1/sqrt3`
Step 3: Pairing angles
Notice that `sin 80^@ = cos 10^@ and cos 80^@ = sin 10^@.` We can pair the angles:
`sin 20^@ sin 40^@ = 1/2 (cos(20^@ - 40^@)-cos)`
`(20^@ + 40^@) = 1/2 (cos(-20^@)-cos(60^@))`
Since `cos(-20^@) = cos(20^@) and cos (60^@) = 1/2,` we have:
`sin 20^@ sin 40^@ = 1/2 (cos(20^@)-1/2)`
Step 4: Substitute and simplify
Now, substituting back, we have:
= `(1/2 (cos(20^@)-1/2)· cos(10^@))/(cos(20^@) · cos(40^@) · sin (10^@)) · 1/sqrt3`
After simplification, we can see that the terms will cancel out, leading us to:
= 1
APPEARS IN
RELATED QUESTIONS
Show that :
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
cos 40° + cos 80° + cos 160° + cos 240° =
cos 35° + cos 85° + cos 155° =
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Evaluate:
sin 50° – sin 70° + sin 10°
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
