Question

1
___________ = cos A sin A
tan A + cot A​

Answer 1

Given :

An equation $\frac{1}{tanA+cotA} =cosAsinA$.

To prove :

That the given expressions are equal.

Step-by-step explanation:

• The expression can be proved by following these steps
• Write $tanA$ in terms of $sinA$ and $cosA$
• We know that,

        $tanA=\frac{sinA}{cosA}$

• Similarly, there is a formula we can use for $cotA$.

        $cotA=\frac{cosA}{sinA}$

• Substitute these values in the left hand side of the equation

        $\frac{1}{ \frac{sinA}{cosA} +\frac{cosA}{sinA} }$

• To make the denominator common, we should decide the least common multiple.

• Take LCM for the denominator

        $\frac{1}{\frac{sin^{2}A+cos^{2}A }{cosAsinA} }$

• Denominator of the denominator becomes numerator
• Take $cosAsinA$ to the numerator.

• So, the equation will become    

         $\frac{cosAsinA}{sin^{2}A+cos^{2}A }$

• We know that, the value of $sin^{2} A+cos^{2} A$ is 1.
• So, the denominator will become

         $\frac{cosAsinA}{1}$    

• Therefore, the final expression would become

        $cosAsinA$

• By the above equations, we proved that the LHS of the equation is equal to the RHS of the equation.

Final answer :

By the above equations we can prove that $\frac{1}{tanA+cotA} =cosAsinA$.