Advertisements
Advertisements
Question
Prove that:
sin 105° + cos 105° = cos 45°
Advertisements
Solution
Consider LHS:
\[\sin 105^\circ + \cos 105^\circ\]
\[ = \sin 105^\circ + \cos \left( 90^\circ + 15^\circ \right)\]
\[ = \sin 105^\circ - \sin 15^\circ\]
\[ = 2\sin \left( \frac{105^\circ - 15^\circ}{2} \right) \cos \left( \frac{105^\circ + 15^\circ}{2} \right) \left\{ \because \sin A + \sin B = 2\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right) \right\}\]
\[ = 2\sin 45^\circ\cos 60^\circ\]
\[ = 2\sin \left( 90^\circ - 45^\circ \right) \cos 60^\circ\]
\[ = 2 \times \frac{1}{2}\cos\left( 45^\circ \right)\]
\[ = \cos 45^\circ\]
Hence, LHS = RHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
Prove that:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
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.
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
cos 40° + cos 80° + cos 160° + cos 240° =
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
cos 35° + cos 85° + cos 155° =
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Evaluate:
sin 50° – sin 70° + sin 10°
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
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.
