Mathematics · Introduction to Trigonometry · NCERT Examples

Example 10

A complete, independently verified solution for NCERT Example 10.

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Mathematics · Chapter 8 · NCERT Examples

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NCERT Example

Example 10

trigonometric identity · proof

Prove that sec A(1 − sin A)(sec A + tan A) = 1.

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

Solution

Board-exam working:

Start with the left-hand side:

LHS=sec⁡A(1−sin⁡A)(sec⁡A+tan⁡A)\mathrm{LHS}=\sec A(1-\sin A)(\sec A+\tan A)
=1cos⁡A(1−sin⁡A)(1cos⁡A+sin⁡Acos⁡A)=\frac1{\cos A}(1-\sin A)\left(\frac1{\cos A}+\frac{\sin A}{\cos A}\right)
=(1−sin⁡A)(1+sin⁡A)cos⁡2A=\frac{(1-\sin A)(1+\sin A)}{\cos^2A}
=1−sin⁡2Acos⁡2A=\frac{1-\sin^2A}{\cos^2A}
=cos⁡2Acos⁡2A=1=RHS=\frac{\cos^2A}{\cos^2A}=1=\mathrm{RHS}

Final answer

Hence proved.

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