Question 986288
If tan theta=cot<60`+theta> find the value of theta

                        SOLUTION

tan&#920; = cot(60 + &#920;) , Find the value of &#920;

Note: tan&#920; = sin&#920;/cos&#920; 
And Also Note:  cot&#920; = cos&#920;/sin&#920;

Hence, tan&#920; = cot(60 + &#920;)

&#8756;(sin&#920;/cos&#920;) = [cos(60 + &#920;)/sin(60 + &#920;)]

 By equating numerator with numerator and denominator with denominator,  we get. .

           &#8756;  sin&#920; = cos(60 + &#920;) 

From trigonometry identity:sin&#920; = cos(90 - &#920;)

By comparing the equation,  we got. ..

              cos(60 + &#920;) = cos(90 - &#920;)

              Council out cos,  we will get 

           &#8756; 60 + &#920; = 90 - &#920;

              Collect the like term

             &#8756; &#920; + &#920; = 90 - 60

              &#8756; 2&#920; = 30

             Divide both side by 2

             &#8756;  &#920; = 30/2

              &#8756; &#920; = 15 °

By equating the denominator, we will have

              Cos&#920; = sin(60 + &#920;)


From trigonometry identity: cos&#920; = sin(90 - &#920;)

   By comparing the equation, we get

            sin(60 + &#920;) = sin(90 - &#920;)

           Cos will council out, we get

                   60 + &#920; = 90 - &#920;

                Collect the like term

                &#8756; &#920; + &#920; = 90 - 60

                &#8756; 2&#920; = 30

                Divide both side by 2

                 &#8756;  &#920; = 30/2

                  &#8756; &#920; = 15 °

Therefore the value of theta , &#920; = 15°

Checking:

tan&#920; = cot(60 + &#920;)

tan15° = cot(60 + 15°)

tan15° = cot75°

Note: cot&#920; = 1/tan&#920;

tan15° = 1/tan75°

0.267949192 = 1/(3.732050808)

0.267949192 = 0.267949192

Therefore, the value of theta , &#920; = 15°