Question

(vií) sing tanſ + sin cosz = 2sin-
$sin \frac{\pi}{3} tan \frac{\pi}{6 } + sin \frac{\pi}{2}cos \frac{\pi}{3 } = 2 {sin}^{2} \frac{\pi}{4}$
​

Answer 1

Correct question :

• $\sf{sin \dfrac{\pi}{3} tan \dfrac{\pi}{6 } + sin \dfrac{\pi}{2}cos \dfrac{\pi}{3 } = 2 {sin}^{2} \dfrac{\pi}{4}}$

Solution :

π is considered to be 180

$\underline{\boxed{ \sf \blue{LHS}}} = \sf \red {sin \dfrac{\pi}{3} tan \dfrac{\pi}{6 } + sin \dfrac{\pi}{2}cos \dfrac{\pi}{3 }} \\ \\ \\ \implies \purple{\sf{ \dfrac{ \sqrt{3} }{2} } \times \dfrac{1}{ \sqrt{3} } + 1 \times \dfrac{1}{2}} \\ \\ \\ \implies \pink{\dfrac{1}{2} + \dfrac{1}{2} = 1} \\ \\ \\ \implies \green{\sf{2 \times \dfrac{1}{2}}} \\ \\ \\ \implies \orange{2 \times (\dfrac{1}{ \sqrt{2} })^{2}} \\ \\ \\ \implies \underline{ \boxed{\gray{\sf{2 \sin^{2} \dfrac{ \pi}{4}}}}} \: \: \: \: \: \: \: \underline{\boxed{\blue{\sf{RHS}}}}$

Addition information :

• tanø = sinø/cosø

• secø = 1/cosø

• cotø = 1/tanø = sinø/cosø

• 1 - tan(ø/2)/1 - tan(ø/2) = ±√1 - sinø/1 + sinø

• tan ø/2 = ±√1 - cosø/1 + cosø

• sinø = Cos(90° - ø)

• cos6= sin(90° - ø)

• tanø = cot(90° - ø)

• cotø = tan(90° - ø)

• secø = cosec(90° - ø)

Answer 2

Answer:

$\underline{\bigstar\:\textsf{According to the given Question :}}$

$:\implies\sf sin \dfrac{\pi}{3} tan \dfrac{\pi}{6 } + sin \dfrac{\pi}{2}cos \dfrac{\pi}{3 } = 2 {sin}^{2} \dfrac{\pi}{4}\\\\\\:\implies\sf sin \dfrac{180}{3} \times tan \dfrac{180}{6 } + sin \dfrac{180}{2}\times cos \dfrac{180}{3 } = 2 {sin}^2 \dfrac{180}{4}\\\\\\:\implies\sf sin(60) \times tan(30) + sin(90) \times cos(60) = 2sin^2(45)\\\\\\:\implies\sf \bigg\lgroup\dfrac{ \sqrt{3} }{2} \times \dfrac{1}{ \sqrt{3}}\bigg\rgroup + \bigg\lgroup1 \times \dfrac{1}{2} \bigg\rgroup = 2 \times \bigg\lgroup \dfrac{1}{ \sqrt{2}} \bigg\rgroup^{2}\\\\\\:\implies\sf \dfrac{1}{2} + \dfrac{1}{2} = 2 \times \dfrac{1}{2}\\\\\\:\implies\sf 1 = 1 \qquad\bigg\lgroup LHS=RHS\bigg\rgroup$

$\rule{180}{1.5}$

$\bigstar\:\sf Trigonometric\:Values :\\\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}& Not D \hat{e} fined \\\cline{1-6}\end{tabular}$