Advertisements
Advertisements
Question
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
Options
1
4
2
None of these
Advertisements
Solution
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to 2.
Explanation:
sinθ + cosecθ = 2
Squaring L.H.S and R.H.S
We get,
⇒ (sinθ + cosecθ)2 = 22
⇒ (sinθ + cosecθ)2 = 4
⇒ sin2θ + cosec2θ + 2sinθ cosecθ = 4 [∵ `1/sintheta = cosectheta`]
= sin2θ + cosec2θ + 2 = 4
= sin2θ + cosec2θ = 2
APPEARS IN
RELATED QUESTIONS
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that
Prove that
If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
If sin α + sin β = a and cos α + cos β = b, show that
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
Write the maximum value of 12 sin x − 9 sin2 x.
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If sinx + cosx = a, then sin6x + cos6x = ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
