NCERT Example 12 — Introduction to Trigonometry

NCERT Example

Example 12

trigonometric identity · secant tangent

Prove that (sin θ − cos θ + 1)/(sin θ + cos θ − 1) = 1/(sec θ − tan θ), using sec²θ = 1 + tan²θ.

Identity using sec θ and tan θ — explanatory sketch (not to scale).

Board-exam working:

Divide the numerator and denominator of the left-hand side by cos θ:

\mathrm{LHS}=\frac{\tan\theta-1+\sec\theta}{\tan\theta+1-\sec\theta}

=\frac{(\tan\theta+\sec\theta)-1}{(\tan\theta-\sec\theta)+1}

Multiply numerator and denominator by tan θ − sec θ:

=\frac{\{(\tan\theta+\sec\theta)-1\}(\tan\theta-\sec\theta)}{\{(\tan\theta-\sec\theta)+1\}(\tan\theta-\sec\theta)}

=\frac{\tan^2\theta-\sec^2\theta-\tan\theta+\sec\theta}{(\tan\theta-\sec\theta+1)(\tan\theta-\sec\theta)}

=\frac{-1-\tan\theta+\sec\theta}{(\tan\theta-\sec\theta+1)(\tan\theta-\sec\theta)}

=\frac{-1}{\tan\theta-\sec\theta}=\frac1{\sec\theta-\tan\theta}=\mathrm{RHS}

Hence proved.