If cosα=35 and cosβ=513I)sin2(α−β2)P)6465II)cos2(α−β2)Q)5665III) tan2(α−β2)R)165IV)sin(α+β)S)164

cosα=35,sinα=45,cosβ=513,sinβ=1213 cos(α-β)=35513+451213=6365 (i) sin2(α−β2)=1−cos(α−β)2=1−63652=165 (ii) cos2(α−β2)=1+cos(α−β)2=1+63652=6465 (iii) tan2(α−β2)=1−cos(α−β)1+cos(α−β)=1−63651+6365=2128=164 (iv) sin(α+β)=45.513+35.1213=5665

If cosα=35 and cosβ=513I)sin2(α−β2)P)6465II)cos2(α−β2)Q)5665III) tan2(α−β2)R)165IV)sin(α+β)S)164

cosα=35,sinα=45,cosβ=513,sinβ=1213 cos(α-β)=35513+451213=6365 (i) sin2(α−β2)=1−cos(α−β)2=1−63652=165 (ii) cos2(α−β2)=1+cos(α−β)2=1+63652=6465 (iii) tan2(α−β2)=1−cos(α−β)1+cos(α−β)=1−63651+6365=2128=164 (iv) sin(α+β)=45.513+35.1213=5665

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If $\cos α = \frac{3}{5}$ and $\cos β = \frac{5}{13}$
I) $\sin^{2} (\frac{α - β}{2})$ P) $\frac{64}{65}$
II) $\cos^{2} (\frac{α - β}{2})$ Q) $\frac{56}{65}$
III) $\tan^{2} (\frac{α - β}{2})$ R) $\frac{1}{65}$
IV) $\sin (α + β)$ S) $\frac{1}{64}$

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Detailed Solution

$\cos α = \frac{3}{5}, \sin α = \frac{4}{5}, \cos β = \frac{5}{13}, \sin β = \frac{12}{13}$
$\cos (α - β) = \frac{3}{5} \frac{5}{13} + \frac{4}{5} \frac{12}{13} = \frac{63}{65}$
(i) $\sin^{2} (\frac{α - β}{2}) = \frac{1 - \cos (α - β)}{2} = \frac{1 - \frac{63}{65}}{2} = \frac{1}{65}$
(ii) $\cos^{2} (\frac{α - β}{2}) = \frac{1 + \cos (α - β)}{2} = \frac{1 + \frac{63}{65}}{2} = \frac{64}{65}$
(iii) $\tan^{2} (\frac{α - β}{2}) = \frac{1 - \cos (α - β)}{1 + \cos (α - β)} = \frac{1 - \frac{63}{65}}{1 + \frac{63}{65}} = \frac{2}{128} = \frac{1}{64}$
(iv) $\sin (α + β) = \frac{4}{5} . \frac{5}{13} + \frac{3}{5} . \frac{12}{13} = \frac{56}{65}$

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I)	$\sin^{2} (\frac{α - β}{2})$	P)	$\frac{64}{65}$
II)	$\cos^{2} (\frac{α - β}{2})$	Q)	$\frac{56}{65}$
III)	$\tan^{2} (\frac{α - β}{2})$	R)	$\frac{1}{65}$
IV)	$\sin (α + β)$	S)	$\frac{1}{64}$

If cosα=35 and cosβ=513I)sin2(α−β2)P)6465II)cos2(α−β2)Q)5665III) tan2(α−β2)R)165IV)sin(α+β)S)164

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If $\cos α = \frac{3}{5}$ and $\cos β = \frac{5}{13}$
I) $\sin^{2} (\frac{α - β}{2})$ P) $\frac{64}{65}$
II) $\cos^{2} (\frac{α - β}{2})$ Q) $\frac{56}{65}$
III) $\tan^{2} (\frac{α - β}{2})$ R) $\frac{1}{65}$
IV) $\sin (α + β)$ S) $\frac{1}{64}$