Chapter 6 – Trigonometry – Text Book Solution
Problem set 6| Q 5.8| Page 138
Prove the following. (tan(theta) / (sec(theta) + 1)) = ((sec(theta) – 1) / tan(theta))
\[\frac{\tan\theta}{\sec\theta + 1}\]
\[ = \frac{\tan\theta}{\sec\theta + 1} \times \frac{\sec\theta - 1}{\sec\theta - 1}\]
\[ = \frac{\tan\theta\left( \sec\theta - 1 \right)}{\sec^2 \theta - 1}\]
\[ = \frac{\tan\theta\left( \sec\theta - 1 \right)}{\tan^2 \theta} \left( 1 + \tan^2 \theta = \sec^2 \theta \right)\]
\[ = \frac{\sec\theta - 1}{\tan\theta}\]
Solution
To prove the given identity: First, let’s write the left-hand side (LHS) of the equation as a single fraction by multiplying the numerator and denominator by cos(theta):
LHS = (sin(theta) / cos(theta)) / (1/cos(theta) + 1) = sin(theta) / (1 + cos(theta))
Next, let’s write the right-hand side (RHS) of the equation as a single fraction by multiplying the numerator and denominator by cos(theta):
RHS = (cos(theta) / sin(theta)) / (cos(theta)/sin(theta) – 1) = cos(theta) / (1 – sin(theta)/cos(theta))
Now, we can simplify both sides by using the trigonometric identity sin^2(theta) + cos^2(theta) = 1:
LHS = sin(theta) / (1 + cos(theta)) = (sin(theta) * (1 – cos(theta))) / (1 – cos^2(theta)) = (sin(theta) * (1 – cos(theta))) / sin^2(theta) = (1 – cos(theta)) / sin(theta)
RHS = cos(theta) / (1 – sin(theta)/cos(theta)) = (cos^2(theta)) / (cos(theta) – sin(theta)) = (1 – sin^2(theta)) / (cos(theta) – sin(theta)) = (1 – sin(theta)) / cos(theta)
Now, we can see that LHS = RHS, and the identity is proven.
Chapter 6 – Trigonometry – Text Book Solution
Problem Set 6 |Q 5.8| P 138
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