Question

Evaluate sin^2 75 -sin^2 45

Answer 1

Answer:

$\boxed{ \sf\huge \implies: \frac{ \sqrt{3} }{4} }$

Step-by-step explanation:

$\sf\ {sin}^{2}75 \degree - {sin}^{2}45 \degree \\ \\ \sf\ \implies: {(sin \: 75 \degree})^{2} - {sin}^{2}45 \degree \\ \\ \sf\ \implies: [{sin(30 \degree + 45 \degree)}^{2}] - {sin}^{2}45 \degree \\ \\ \tt\underline{we \: know \bf\red{{\:sin(A+B)=sin(A)cos(B)+cos(A)sin(B) }}} \\ \\\sf\ \implies:{(sin30 \degree \: cos45 + \degree \: cos30 \degree \: sin45 \degree})^{2} - {sin}^{2}45 \degree \\ \\ \sf\ \implies:( { \frac{1}{2 } \times \frac{1 }{ \sqrt{2} } + \frac{ \sqrt{3} }{2} \times \frac{1}{ \sqrt{2} } })^{2} - {( \frac{1}{ \sqrt{2} }})^{2} \\ \\\sf\ \implies: {(\frac{ 1 + \sqrt{3} }{2 \sqrt{2} }})^{2} - \frac{1}{2} \\ \\ \sf\ \implies: \frac{ {(1 + \sqrt{3} })^{2} }{8} - \frac{1}{2} \\ \\ \sf\ \implies: \frac{ {(1 + \sqrt{3} })^{2} - 4}{8} \\ \\\sf\ \implies: \frac{1 + 3 +2 \sqrt{3} - 4 }{8} \\ \\ \sf\ \implies: \frac{2 \sqrt{3} }{8} \\ \\ \boxed{ \sf\huge \implies: \frac{ \sqrt{3} }{4} }$