Advertisements
Advertisements
प्रश्न
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
Advertisements
उत्तर
LH.S = `(1 - cos2"A")/2 + (1 - cos2"B")/2 - (1 - cos2"C")/2`
Hint: `[sin^2"A" = (1 - cos2"A")/2]`
= `[1/2 + 1/2 - 1/2] - 1/2 [cos2"A" + cos2"B" - cos 2"C"]`
= `1/2 - 1/2 [2cos("A" + "B") cos("A" - "B") - (2cos^2"C" - 1)]`
= `1/2 - cos("A" + "B") cos("A" - "B") + cos^2"C" - 1/2`
= cos C cos(A – B) + cos2C
= cos C [cos(A – B) – cos(A + B)]
= cos C [2 sin A sin B]
= 2 sin A sin B cos C
= R.H.S
APPEARS IN
संबंधित प्रश्न
Find all the angles between 0° and 360° which satisfy the equation sin2θ = `3/4`
Find the value of cos 105°.
Prove that cos(π + θ) = − cos θ
Prove that sin(π + θ) = − sin θ.
Prove that sin 105° + cos 105° = cos 45°
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B
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
Prove that cot(A + B) = `(cot "A" cot "B" - 1)/(cot "A" + cot "B")`
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
If θ is an acute angle, then find `cos (pi/4 + theta/2)`, when sin θ = `8/9`
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
Express the following as a sum or difference
2 sin 10θ cos 2θ
Prove that cos(30° – A) cos(30° + A) + cos(45° – A) cos(45° + A) = `cos 2"A" + 1/4`
If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin2 B + sin2 C = 1
Choose the correct alternative:
`(1 + cos pi/8) (1 + cos (3pi)/8) (1 + cos (5pi)/8) (1 + cos (7pi)/8)` =
Choose the correct alternative:
Let fk(x) = `1/"k" [sin^"k" x + cos^"k" x]` where x ∈ R and k ≥ 1. Then f4(x) − f6(x) =
