Advertisements
Advertisements
प्रश्न
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Advertisements
उत्तर
\[\frac{1}{\sin A} + \frac{1}{\cos A} = \frac{1}{\sin B} + \frac{1}{\cos B}\]
\[ \Rightarrow \frac{1}{\sin A} - \frac{1}{\sin B} = \frac{1}{\cos B} - \frac{1}{\cos A}\]
\[ \Rightarrow \frac{\sin B - \sin A}{\sin A\sin B} = \frac{\cos A - \cos B}{\cos A\cos B}\]
\[ \Rightarrow \frac{\sin B - \sin A}{\cos A - \cos B} = \frac{\sin A\sin B}{\cos A\cos B}\]
\[ \Rightarrow \frac{2\sin\left( \frac{B - A}{2} \right)\cos\left( \frac{A + B}{2} \right)}{- 2\sin\left( \frac{A - B}{2} \right)\sin\left( \frac{A + B}{2} \right)} = \frac{\sin A\sin B}{\cos A\cos B}\]
\[ \Rightarrow \frac{- \sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)}{- \sin\left( \frac{A - B}{2} \right)\sin\left( \frac{A + B}{2} \right)} = \frac{\sin A\sin B}{\cos A\cos B}\]
\[ \Rightarrow \frac{\cos\left( \frac{A + B}{2} \right)}{\sin\left( \frac{A + B}{2} \right)} = \frac{\sin A\sin B}{\cos A\cos B}\]
\[ \Rightarrow \cot\left( \frac{A + B}{2} \right) = \tan A\tan B\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
The value of sin 50° − sin 70° + sin 10° is equal to
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
`cos (7"A")/3 sin (5"A")/3`
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
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
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
