Question

sin 75° + cos²15° = ?​

Answer 1

using identity sin(90-theta)=cos theta

and use sin theta+cos theta =1...

Answer 2

we know that

$cos2x = 2 {cos}^{2} x - 1 \\ {cos}^{2} x = \frac{cos2x + 1}{2}$

and

$sin(x + y) = sinxcosy + cosxsiny$

now,

$sin {75}^{o} + {cos}^{2} {15}^{o} \\ = sin( {30}^{o} + {45}^{o} ) + \frac{cos {(2 \times 15)}^{o} + 1}{2} \\ = sin {30}^{o} cos {45}^{o} + cos {30}^{o} sin {45}^{o} \\ \: \: \: \: + \frac{cos {30}^{o} + 1 }{2} \\ = ( \frac{1}{2})( \frac{1}{ \sqrt{2} } ) + ( \frac{ \sqrt{3} }{2} )( \frac{1}{ \sqrt{2} } ) + \frac{ \frac{ \sqrt{3} }{2} + 1}{2} \\ = \frac{ \sqrt{3} + 1}{2 \sqrt{2} } + \frac{ \frac{ \sqrt{3} + 2}{2} }{2} \\ = \frac{( \sqrt{3} + 1) \sqrt{2} }{2 \sqrt{2} \sqrt{2} } + \frac{ \sqrt{3} + 2 }{4} \\ = \frac{( \sqrt{3} + 1) \sqrt{2} + ( \sqrt{3} + 1) + 1}{4} \\ = \frac{( \sqrt{3} + 1)( \sqrt{2} + 1) + 1 }{4}$