Advertisements
Advertisements
Question
If A + B + C = 2s, then prove that sin(s – A) sin(s – B)+ sin s sin(s – C) = sin A sin B
Solution
Now sin(s – A) sin(s – B) = `1/2 {cos[("s" - "A") - ("s" - "B")] - cos[("s" - "A") + ("s" - "B")]}`
= `1/2cos("s" - "A" - "s" + "B") - cos[2"s" - ("A" + "B")]`
= `1/2 {cos("A" - "B) - cos"C"}` .....[∴ cos(A – B) = cos(B – A)]
Again sin s sin s – C = `1/2[cos"C" - cos("A" + "B")`
So, L.H.S = `1/2 {cos("A" - "B") - cos"C" + cos"C" - cos("A" + "B")}`
= `1/2 [cos("A" - "B") - cos("A" + "B")`
= `1/2 [2sin"A" sin"B"]`
= sin A sin B
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of `tan ((19pi)/3)`
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`
Prove that sin(A + B) sin(A – B) = sin2A – sin2B
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
If cos(α – β) + cos(β – γ) + cos(γ – α) = `- 3/2`, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0
Show that tan(45° − A) = `(1 - tan "A")/(1 + tan "A")`
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
Express the following as a sum or difference
cos 5θ cos 2θ
Express the following as a sum or difference
sin 5θ sin 4θ
Express the following as a product
cos 65° + cos 15°
Show that `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` = tan 2x
Show that cot(A + 15°) – tan(A – 15°) = `(4cos2"A")/(1 + 2 sin2"A")`
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
