Solve frac{1-sin^{2}30^{circ}}{1+sin^{2}45^{circ}}*frac{cos^{2}60^{circ}+cos^{2}30^{circ}}{csc^{2}90^{circ}-cot^290^circ}+(sin60^circtan30^circ)

$$\frac { 1 - \sin ^ { 2 } 30 ^ { \circ } } { 1 + \sin ^ { 2 } 45 ^ { \circ } } \times \frac { \cos ^ { 2 } 60 ^ { \circ } + \cos ^ { 2 } 30 ^ { \circ } } { \csc ^ { 2 } 90 ^ { \circ } - \cot ^ { 2 } 90 ^ { \circ } } + ( \sin 60 ^ { \circ } \tan 30 ^ { \circ } )$$

Evaluate

$$1$$

Short Solution Steps

Get the value of \sin(30) from trigonometric values table.

$$\frac{1-\left(\frac{1}{2}\right)^{2}}{1+\left(\sin(45)\right)^{2}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1-\frac{1}{4}}{1+\left(\sin(45)\right)^{2}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

$$\frac{\frac{3}{4}}{1+\left(\sin(45)\right)^{2}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Get the value of \sin(45) from trigonometric values table.

$$\frac{\frac{3}{4}}{1+\left(\frac{\sqrt{2}}{2}\right)^{2}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

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

$$\frac{\frac{3}{4}}{1+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2^{2}}{2^{2}}.

$$\frac{\frac{3}{4}}{\frac{2^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Since \frac{2^{2}}{2^{2}} and \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3}{4}}{\frac{2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Divide \frac{3}{4} by \frac{2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}} by multiplying \frac{3}{4} by the reciprocal of \frac{2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}.

$$\frac{3\times 2^{2}}{4\left(2^{2}+\left(\sqrt{2}\right)^{2}\right)}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3\times 4}{4\left(2^{2}+\left(\sqrt{2}\right)^{2}\right)}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Multiply 3 and 4 to get 12.

$$\frac{12}{4\left(2^{2}+\left(\sqrt{2}\right)^{2}\right)}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12}{4\left(4+\left(\sqrt{2}\right)^{2}\right)}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

The square of \sqrt{2} is 2.

$$\frac{12}{4\left(4+2\right)}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Add 4 and 2 to get 6.

$$\frac{12}{4\times 6}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Multiply 4 and 6 to get 24.

$$\frac{12}{24}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Reduce the fraction \frac{12}{24} to lowest terms by extracting and canceling out 12.

$$\frac{1}{2}\times \frac{\left(\cos(60)\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{1}{2}\times \frac{\left(\frac{1}{2}\right)^{2}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{2}\times \frac{\frac{1}{4}+\left(\cos(30)\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Get the value of \cos(30) from trigonometric values table.

$$\frac{1}{2}\times \frac{\frac{1}{4}+\left(\frac{\sqrt{3}}{2}\right)^{2}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{2}\times \frac{\frac{1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{1}{2}\times \frac{\frac{1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{4}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Since \frac{1}{4} and \frac{\left(\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\times \frac{\frac{1+\left(\sqrt{3}\right)^{2}}{4}}{\left(\csc(90)\right)^{2}-\left(\cot(90)\right)^{2}}+\sin(60)\tan(30)$$

Get the value of \cot(90) from trigonometric values table.

$$\frac{1}{2}\times \frac{\frac{1+\left(\sqrt{3}\right)^{2}}{4}}{\left(\csc(90)\right)^{2}-0^{2}}+\sin(60)\tan(30)$$

Calculate 0 to the power of 2 and get 0.

$$\frac{1}{2}\times \frac{\frac{1+\left(\sqrt{3}\right)^{2}}{4}}{\left(\csc(90)\right)^{2}-0}+\sin(60)\tan(30)$$

The square of \sqrt{3} is 3.

$$\frac{1}{2}\times \frac{\frac{1+3}{4}}{\left(\csc(90)\right)^{2}-0}+\sin(60)\tan(30)$$

Add 1 and 3 to get 4.

$$\frac{1}{2}\times \frac{\frac{4}{4}}{\left(\csc(90)\right)^{2}-0}+\sin(60)\tan(30)$$

Divide 4 by 4 to get 1.

$$\frac{1}{2}\times \frac{1}{\left(\csc(90)\right)^{2}-0}+\sin(60)\tan(30)$$

Multiply \frac{1}{2} times \frac{1}{\left(\csc(90)\right)^{2}-0} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{2\left(\left(\csc(90)\right)^{2}-0\right)}+\sin(60)\tan(30)$$

Get the value of \sin(60) from trigonometric values table.

$$\frac{1}{2\left(\left(\csc(90)\right)^{2}-0\right)}+\frac{\sqrt{3}}{2}\tan(30)$$

Get the value of \tan(30) from trigonometric values table.

$$\frac{1}{2\left(\left(\csc(90)\right)^{2}-0\right)}+\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{3}$$

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

$$\frac{1}{2\left(\left(\csc(90)\right)^{2}-0\right)}+\frac{\sqrt{3}\sqrt{3}}{2\times 3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(\left(\csc(90)\right)^{2}-0\right) and 2\times 3 is 2\times 3. Multiply \frac{1}{2\left(\left(\csc(90)\right)^{2}-0\right)} times \frac{3}{3}.

$$\frac{3}{2\times 3}+\frac{\sqrt{3}\sqrt{3}}{2\times 3}$$

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

$$\frac{3+\sqrt{3}\sqrt{3}}{2\times 3}$$

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

$$\frac{3+3}{2\times 3}$$

Do the calculations in 3+3.

$$\frac{6}{2\times 3}$$

Expand 2\times 3.

$$\frac{6}{6}$$

Divide 6 by 6 to get 1.

$$1$$