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Question: If \(\tan A = \cot B\), Prove that \(A + B = 90^\circ \)...

If tan⁡A=cot⁡B\tan A = \cot B, Prove that A+B=90∘A + B = 90^\circ

Explanation

Solution

If an expression connects two trigonometric functions, find the relation between the angles given in the expression. The relation thus obtained is converted into a trigonometric function by applying the trigonometric identity and thus prove the given equation.

Complete step-by-step answer:
Consider the given expression.
tan⁡A=cot⁡B\tan A = \cot B
The tangent of an angle is the ratio of the sine of the angle to the cosine of the same angle, and the cotangent of an angle is the ratio of the cosine of the angle to the sine of the same angle,
Apply the above relation on the given expression to get,
tan⁡A=cot⁡B sin⁡Acos⁡A=cos⁡Bsin⁡B  \tan A = \cot B \\\ \dfrac{{\sin A}}{{\cos A}} = \dfrac{{\cos B}}{{\sin B}} \\\
Now, cross multiply the above relation and bring all the terms into one side to get,

sin⁡Asin⁡B=cos⁡Acos⁡B cos⁡Acos⁡B−sin⁡Asin⁡B=0 \sin A\sin B = \cos A\cos B \\\ \cos A\cos B - \sin A\sin B = 0

The above relation is an expression connecting sine and cosine angles. Thus, the above relation is an identity of a trigonometric function, the trigonometric identity connecting the above relation is,
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A\cos B - \sin A\sin B
Now, replace the given expression with the above trigonometric relation to getting,
cos⁡(A+B)=0\cos (A + B) = 0
Now, make the term to in such a way that the required value (A+B)\left( {A + B} \right) on one side and the other terms into the other sides to get,
A+B=cos⁡−1(0)A + B = {\cos ^{ - 1}}(0)
The inverse trigonometric value of cosine zero is equal to 90 degrees. Now apply the value of the inverse value of the cosine zero to get the required result.
A+B=cos⁡−1(0) =90∘  A + B = {\cos ^{ - 1}}(0) \\\ = 90^\circ \\\
Hence if tan⁡A=cot⁡B\tan A = \cot B then the sum of the angles of the tangent and cotangent is,
A+B=90∘A + B = 90^\circ
Hence the given relation is proved.

Note: When an angle of a trigonometric function is related to the other angle of a trigonometric function, then the relation between the trigonometric function formula is used to obtain the required result.Students should remember the important trigonometric formulas,identities and standard trigonometric angles for solving these types of problems.