Question

Sin^2 1°+sin^2 3°+sin^2 5°+..........+sin^2 85°+sin87°^2+sin^2 89°

Answer 1

Answer:

$\bold\red{value=\frac{45}{2}}$

Step-by-step explanation:

Given,

${ \sin }^{2} 1 + { \sin }^{2} 3+ ...... + { \sin }^{2} 87 + { \sin}^{2} 89$

Now ,

we know that,

$\sin( \alpha ) = \cos(90 - \alpha )$

Therefore,

we get

$= { \sin}^{2} 1 + { \sin}^{2} 3 + ....... + { \cos }^{2}3 + { \cos }^{2} 1 \\ \\ = ( { \sin }^{2} 1 + { \cos }^{2} 1) + ( { \sin }^{2}3 + { \cos }^{2} 3) + ..... + { \sin }^{2} 45$

But,

we know that,

${ \sin }^{2} \alpha + { \cos}^{2} \alpha = 1$

Therefore,

we get,

$= 1 + 1 + ......... \: upto \: 22 \: terms + { \sin}^{2} 45$

But,

$\sin(45) = \frac{1}{ \sqrt{2} } \\ \\ = > { \sin }^{2} 45 = \frac{1}{2}$

Putting the values,

we get,

$= 1 \times 22 + \frac{1}{2} \\ \\ = 22 + \frac{1}{2} \\ \\ = \frac{45}{2}$

Hence,

$\bold{value=\frac{45}{2}}$