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Q(25)  Prove the following

\small \cos6x = 32\cos^{6}x -48\cos^{4}x + 18\cos^{2}x-1

We know that  cos 3x = 4 - 3cos x we use this in our problem we can write cos 6x as cos 3(2x)         cos 3(2x) = 4 - 3 cos 2x                          =    -                                                                                                                   =                                            = 32 - 4 - 48 + 24  -                           =   32  - 48 + 18 - 1  = R.H.S.

Q(24) Prove the following 

\small \cos4x = 1 - 8\sin^{2}x\cos^{2}x

We know that                    We use this in our problem   cos 4x = cos 2(2x)             =                =                                                      =  = R.H.S.

Q(23) Prove that

 \small \tan4x = \frac{4\tan x(1-\tan^{2}x)}{1-6 \tan^{2}x+\tan^{4}x}

We know that       and we can write tan 4x = tan 2(2x) So,       =                                                              =                                                      =                                                       =           = R.H.S.

prove the following

Q(22) \small \cot x \cot2x - \cot2x\cot3x - \cot3x\cot x =1

cot x cot2x - cot3x(cot2x - cotx) Now we can write cot3x = cot(2x + x) and we know that  So,                                 =   cotx cot2x - (cot2xcotx -1)            =  cotx cot2x - cot2xcotx +1             = 1  = R.H.S.

Q (21)  Prove the following

\small \frac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x

We know that  We use these identities                     =RHS

Q (20)  Prove the following

\small \frac{\sin x - \sin 3x}{\sin^{2}x-\cos^{2}x} = 2\sin x

We know that   We use these  identities                                                                                                         R.H.S.

Q (18) Prove the following

\small \frac{\sin x - \sin y}{\cos x+\cos y} = \tan \frac{(x-y)}{2}

We know that We use these identities  R.H.S.

Q (17) Prove the following

\small \frac{\sin5x + \sin3x}{\cos5x + \cos3x} = \tan4x

We know that We use these identities                                                                                                           R.H.S.

Q (16) Prove the following

\small \frac{\cos 9x - \cos 5x}{\sin17x - \sin3x} = -\frac{\sin2x}{\cos10x}

As we know that                                                                                                         R.H.S.

Q (15) Prove the following

\small \cot4x(\sin5x + \sin3x) = \cot x(\sin5x - \sin3x)

We know that            By using this , we get  sin5x + sin3x = 2sin4xcosx now nultiply and divide by sin x Now we know that By using this our equation becomes                                                       R.H.S.

Q (14) Prove the following

\small \sin2x +2\sin4x + \sin6x = 4\cos^{2}x\sin4x

We know that  We are using this identity  sin2x + 2sin4x + sin6x = (sin2x + sin6x) + 2sin4x  sin2x + sin6x = 2sin4xcos(-2x) =  2sin4xcos(2x)          ( cos(-x) = cos x) So, our equation becomes sin2x + 2sin4x + sin6x = 2sin4xcos(2x) + 2sin4x Now, take the 2sin4x common sin2x + 2sin4x + sin6x = 2sin4x(cos2x +1)           (   )                                      =2sin4x( +1 )      ...

Q (13) Prove the following

\small \cos^{2}2x - \cos^{2}6x = \sin4x\sin8x

As we know that    Now         By using these identities cos2x - cos6x = -2sin(4x)sin(-2x) = 2sin4xsin2x                  (  sin(-x) = -sin x                                                                                                     cos(-x) = cosx) cos2x + cos 6x = 2cos4xcos(-2x) = 2cos4xcos2x So our equation becomes                                                     ...

Q (12) Prove the following

\small \sin^{2}6x - \sin^{2}4x = \sin2x\sin10x

We know that  So,  Now,  we know that  By using these identities sin6x + sin4x = 2sin5xcosx sin6x - sin4x = 2cos5xsinx Now,  2sinAcosB = sin(A+B) + sin(A-B) 2cosAsinB = sin(A+B) - sin(A-B) by using these identities 2cos5xsin5x = sin10x - 0 2sinxcosx = sin2x + 0 hence                                                                           

Q (11) Prove the following

\small \cos \left ( \frac{3\pi }{4}+x \right ) - \cos\left ( \frac{3\pi }{4} -x\right ) = -\sqrt{2} \sin x
 

We know that  [ cos(A+B) - cos (A-B) = -2sinAsinB ] By using this identity                                   R.H.S.

Q (10) Prove the following

\small \sin (n+1)x\sin(n+2)x + \cos(n+1)x\cos(n+2)x =\cos x

Multiply and divide by 2  Now by using identities -2sinAsinB = cos(A+B) - cos(A-B) 2cosAcosB =  cos(A+B) + cos(A-B)                                         R.H.S.

Q (9) Prove the following

\small \cos \left ( \frac{3\pi }{2} +x\right )\cos (2\pi +x)\left [ \cot \left ( \frac{3\pi }{2}-x \right ) +\cot (2\pi +x)\right ] = 1

We know that  So, by using these our equation simplifies to R.H.S.

Q (8) Prove the following

\small \frac{\cos (\pi +x)\cos (-x)}{\sin (\pi -x)\cos \left ( \frac{\pi }{2}+x \right )} = \cot ^{2} x

 

As we know that,    ,      ,                 and     By using these our equation simplify to                                                                                                R.H.S.

Q (7)  Prove the following

\small \frac{\tan \left ( \frac{\pi }{4}+x \right )}{\tan \left ( \frac{\pi }{4} -x\right )} = \left ( \frac{1+\tan x}{1-\tan x} \right )^{2}

As we know that      and    So, by using these identities                                                                                                                           R.H.S

Prove the following:

Q(6) \small \cos \left ( \frac{\pi }{4}-x \right )\cos \left ( \frac{\pi }{4}-y \right ) - \sin \left ( \frac{\pi }{4} -x\right )\sin \left ( \frac{\pi }{4}-y \right ) =\sin (x+y)
 

Multiply and divide by 2 both cos and sin functions We get, Now, we know that 2cosAcosB = cos(A+B) + cos(A-B)             -(i) -2sinAsinB = cos(A+B) - cos(A-B)               -(ii)  We use these two identities In our question A =                            B =    So,  As we know that By using this                                                                     ...

Q (5) Find the value of 

\small (ii) \tan 15\degree

We know that,                                                    By using this we can write                                                        
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