Mathematics · Introduction to Trigonometry · NCERT Examples

Example 11

A complete, independently verified solution for NCERT Example 11.

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Mathematics · Chapter 8 · NCERT Examples

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NCERT Example

Example 11

trigonometric identity · proof

Prove that (cot A − cos A)/(cot A + cos A) = (cosec A − 1)/(cosec A + 1).

Identity using cot A and cosec AA right triangle marks angle A, its opposite side, adjacent side and hypotenuse so the trigonometric ratios can be read directly.adjacentoppositehypotenuseAABC
Identity using cot A and cosec A — explanatory sketch (not to scale).

Solution

Board-exam working:

LHS=cot⁡A−cos⁡Acot⁡A+cos⁡A\mathrm{LHS}=\frac{\cot A-\cos A}{\cot A+\cos A}
=cos⁡Asin⁡A−cos⁡Acos⁡Asin⁡A+cos⁡A=\frac{\frac{\cos A}{\sin A}-\cos A}{\frac{\cos A}{\sin A}+\cos A}
=cos⁡A(1sin⁡A−1)cos⁡A(1sin⁡A+1)=\frac{\cos A\left(\frac1{\sin A}-1\right)}{\cos A\left(\frac1{\sin A}+1\right)}
=cosec⁡A−1cosec⁡A+1=RHS=\frac{\cosec A-1}{\cosec A+1}=\mathrm{RHS}

Final answer

Hence proved.

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