Solve sinpi/3tanpi/6+sinpi/2cospi/3=2sin^2frac{pi}{4}

$$\sin \frac { \pi } { 3 } \tan \frac { \pi } { 6 } + \sin \frac { \pi } { 2 } \cos \frac { \pi } { 3 } = 2 \sin ^ { 2 } \frac { \pi } { 4 }$$

Verify

Short Solution Steps

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

$$\frac{\sqrt{3}}{2}\tan(\frac{\pi }{6})+\sin(\frac{\pi }{2})\cos(\frac{\pi }{3})=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Get the value of \tan(\frac{\pi }{6}) from trigonometric values table.

$$\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{3}+\sin(\frac{\pi }{2})\cos(\frac{\pi }{3})=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Multiply \frac{\sqrt{3}}{2} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{3}\sqrt{3}}{2\times 3}+\sin(\frac{\pi }{2})\cos(\frac{\pi }{3})=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Get the value of \sin(\frac{\pi }{2}) from trigonometric values table.

$$\frac{\sqrt{3}\sqrt{3}}{2\times 3}+1\cos(\frac{\pi }{3})=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{\sqrt{3}\sqrt{3}}{2\times 3}+1\times \frac{1}{2}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Multiply 1 and \frac{1}{2} to get \frac{1}{2}.

$$\frac{\sqrt{3}\sqrt{3}}{2\times 3}+\frac{1}{2}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 3 and 2 is 2\times 3. Multiply \frac{1}{2} times \frac{3}{3}.

$$\frac{\sqrt{3}\sqrt{3}}{2\times 3}+\frac{3}{2\times 3}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Since \frac{\sqrt{3}\sqrt{3}}{2\times 3} and \frac{3}{2\times 3} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{3}\sqrt{3}+3}{2\times 3}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Do the multiplications in \sqrt{3}\sqrt{3}+3.

$$\frac{3+3}{2\times 3}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Do the calculations in 3+3.

$$\frac{6}{2\times 3}=2\left(\sin(\frac{\pi }{4})\right)^{2}$$

Get the value of \sin(\frac{\pi }{4}) from trigonometric values table.

$$\frac{6}{2\times 3}=2\times \left(\frac{\sqrt{2}}{2}\right)^{2}$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{6}{2\times 3}=2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Express 2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{6}{2\times 3}=\frac{2\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{6}{2\times 3}=\frac{\left(\sqrt{2}\right)^{2}}{2}$$

Multiply 2 and 3 to get 6.

$$\frac{6}{6}=\frac{\left(\sqrt{2}\right)^{2}}{2}$$

Divide 6 by 6 to get 1.

$$1=\frac{\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$1=\frac{2}{2}$$

Divide 2 by 2 to get 1.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$