Advertisements
Advertisements
Question
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Advertisements
Solution
L.H.S = cos(A + B) cos(A – B)
= (cos A cos B – sin A sin B)(cos A cos B + sin (A sin B)
= cos2A cos2B – sin2A sin2B
= cos2A (1 – sin2B) – (1 – cos2A) sin2B
= cos2A – cos2A sin2B – sin2B + cos2A sin2B
= cos2A – sin2B
= R.H.S
Now cos2A – sin2B = (1 – sin2A) – (1 – cos2B)
= 1 – sin2A – 1 + cos2B
= cos2B – sin2A
APPEARS IN
RELATED QUESTIONS
Find the values of tan(1050°)
Find the values of cot(660°)
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II 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 sin(x + y)
Find the value of cos 105°.
Prove that sin(π + θ) = − sin θ.
Prove that sin(30° + θ) + cos(60° + θ) = cos θ
Show that tan 75° + cot 75° = 4
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Show that tan(45° + A) = `(1 + tan"A")/(1 - tan"A")`
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
Express the following as a product
cos 35° – cos 75°
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
If A + B + C = 180°, prove that `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
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
