TS 10th Class Maths Solutions Chapter 11 Trigonometry Optional Exercise

Students can practice 10th Class Maths Study Material Telangana Chapter 11 Trigonometry Optional Exercise to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 11 Trigonometry Optional Exercise

Question 1.
Prove that \(\frac{\cot \theta-\cos \theta}{\cot \theta+\cos \theta}\) = \(\frac{{cosec} \theta-1}{{cosec} \theta+1}\)
Solution:

(Dividing the numerator and denominator by ‘sin θ’)

Question 2.
Prove that \(\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}\) = \(\frac{1}{\sec \theta-\tan \theta}\) using the identify sec²θ = 1 + tan²θ.
Solution:

Question 3.
Prove that (cosec A – sin A) (sec A – cos A) = \(\frac{1}{\tan A+\cot A}\)
Solution:
L.H.S. = (cosec A – sin A) (sec A – cos A)
= (\(\frac{1}{\sin \mathrm{A}}\) – sin A) (\(\frac{1}{\cos \mathrm{A}}\) – cos A)
(∵ sin²A + cos²A = 1 ⇒ 1 – sin²A = cos²A and 1 – cos²A = sin²A)

∴ L.H.S. = R.H.S
Hence (cosec A – sin A) (sec A – cos A)
= \(\frac{1}{\tan A+\cot A}\)

Question 4.
Prove that \(\frac{1+\sec \mathrm{A}}{\sec \mathrm{A}}\) = \(\frac{\sin ^2 A}{1-\cos A}\)
Solution:

(∵ sin²A + cos²A = 1 ⇒ sin²A = 1 – cos²A)

Question 5.
Show that \(\left(\frac{1+\tan ^2 \mathrm{~A}}{1+\cot ^2 \mathrm{~A}}\right)\) = \(\left(\frac{1+\tan A}{1-\cot A}\right)^2\) = tan²A
Solution:

Question 6.
Prove that \(\frac{(\sec A-1)}{(\sec A+1)}\) = \(\frac{(1-\cos \mathrm{A})}{(1+\cos \mathrm{A})}\)
Solution: