Solve frac{sqrt{3}-tan15^circ}{1+sqrt{3}tan15^circ}

$$\frac { \sqrt { 3 } - \tan 15 ^ { \circ } } { 1 + \sqrt { 3 } \tan 15 ^ { \circ } }$$

Evaluate

$$\frac{107179676972449082242158358245529\sqrt{3}-107179676972449080000000000000000}{78460969082652753273524925263413}\approx 1$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{\sqrt{3} - 0.2679491924311227}{1 + \sqrt{3} 0.2679491924311227}$$

Rationalize the denominator of \frac{\sqrt{3}-0.2679491924311227}{1+0.2679491924311227\sqrt{3}} by multiplying numerator and denominator by 1-0.2679491924311227\sqrt{3}.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{\left(1+0.2679491924311227\sqrt{3}\right)\left(1-0.2679491924311227\sqrt{3}\right)}$$

Consider \left(1+0.2679491924311227\sqrt{3}\right)\left(1-0.2679491924311227\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1^{2}-\left(0.2679491924311227\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1-\left(0.2679491924311227\sqrt{3}\right)^{2}}$$

Expand \left(0.2679491924311227\sqrt{3}\right)^{2}.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1-0.2679491924311227^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 0.2679491924311227 to the power of 2 and get 0.07179676972449082242158358245529.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1-0.07179676972449082242158358245529\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1-0.07179676972449082242158358245529\times 3}$$

Multiply 0.07179676972449082242158358245529 and 3 to get 0.21539030917347246726475074736587.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{1-0.21539030917347246726475074736587}$$

Subtract 0.21539030917347246726475074736587 from 1 to get 0.78460969082652753273524925263413.

$$\frac{\left(\sqrt{3}-0.2679491924311227\right)\left(1-0.2679491924311227\sqrt{3}\right)}{0.78460969082652753273524925263413}$$

Apply the distributive property by multiplying each term of \sqrt{3}-0.2679491924311227 by each term of 1-0.2679491924311227\sqrt{3}.

$$\frac{\sqrt{3}-0.2679491924311227\left(\sqrt{3}\right)^{2}-0.2679491924311227+0.07179676972449082242158358245529\sqrt{3}}{0.78460969082652753273524925263413}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}-0.2679491924311227\times 3-0.2679491924311227+0.07179676972449082242158358245529\sqrt{3}}{0.78460969082652753273524925263413}$$

Multiply -0.2679491924311227 and 3 to get -0.8038475772933681.

$$\frac{\sqrt{3}-0.8038475772933681-0.2679491924311227+0.07179676972449082242158358245529\sqrt{3}}{0.78460969082652753273524925263413}$$

Subtract 0.2679491924311227 from -0.8038475772933681 to get -1.0717967697244908.

$$\frac{\sqrt{3}-1.0717967697244908+0.07179676972449082242158358245529\sqrt{3}}{0.78460969082652753273524925263413}$$

Combine \sqrt{3} and 0.07179676972449082242158358245529\sqrt{3} to get 1.07179676972449082242158358245529\sqrt{3}.

$$\frac{1.07179676972449082242158358245529\sqrt{3}-1.0717967697244908}{0.78460969082652753273524925263413}$$