Advertisements
Advertisements
प्रश्न
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
Advertisements
उत्तर
A + B + C = 180°
Given A = 90°
∴ B + C = 90°
⇒ `("B" + "C")/2` = 45°
`"B"/2 + "C"/2` = 45°
R.H.S = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
= `1 + sqrt(2) (2cos "B"/2 sin "C"/2)`
We know that 2 cosA sinB = sin(A + B) – sin(A – B)
= `- 1 + sqrt(2) (sin (("B" + "C"))/2 - sin (("B" - "C"))/2)`
= `- 1 + sqrt(2) (sin 45^circ - sin (("B" - "C"))/2)`
= `- 1 + sqrt(2) (1/sqrt(2) - sin (("B" - "C")^2)/2)`
= `- 1 + 1 - sqrt(2) sin (("B" - "C"))/2`
= `- sqrt(2) sin (("B" - "C"))/2` .....(1)
L.H.S = cos B – cos C
= `2 sin (("B" + "C"))/2 sin (("C" - "B"))/2`
= `2 sin 45^circ sin (("C" - "B"))/2`
= `2(1/sqrt(2)) sin ((-("B" - "C"))/2)`
= `- sqrt(2) sin (("B" - "C"))/2` .....(2)
From (1) and (2)
⇒ L.H.S = R.H.S
APPEARS IN
संबंधित प्रश्न
Find the values of cot(660°)
Find the values of `tan ((19pi)/3)`
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of tan(x + y)
If sin A = `3/5` and cos B = `9/41, 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of cos(A – B)
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`
Find the value of tan `(7pi)/12`
Find a quadratic equation whose roots are sin 15° and cos 15°
Prove that sin 75° – sin 15° = cos 105° + cos 15°
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4
Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ
Express the following as a sum or difference
sin 35° cos 28°
If A + B + C = 2s, then prove that sin(s – A) sin(s – B)+ sin s sin(s – C) = sin A sin B
If x + y + z = xyz, then prove that `(2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)`
Choose the correct alternative:
`(sin("A" - "B"))/(cos"A" cos"B") + (sin("B" - "C"))/(cos"B" cos"C") + (sin("C" - "A"))/(cos"C" cos"A")` is
