Class 10 Maths · Introduction to Trigonometry · NCERT Exercise 8.3 · Question 4

Introduction to Trigonometry Medium

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(i) $(csc θ - cot θ)^{2} = \frac{1 - cos θ}{1 + cos θ}$ .
(ii) $\frac{cos A}{1 + sin A} + \frac{1 + sin A}{cos A} = 2 sec A$ .
(iii) $\frac{tan θ}{1 - cot θ} + \frac{cot θ}{1 - tan θ} = 1 + sec θ csc θ$ .
(iv) $\frac{1 + sec A}{sec A} = \frac{sin ^{2} A}{1 - cos A}$ .
(v) $\frac{cos A - sin A + 1}{cos A + sin A - 1} = csc A + cot A$ .
(vi) $\frac{1 + sin A}{1 - sin A} = sec A + tan A$ .
(vii) $\frac{sin θ - 2 sin ^{3} θ}{2 cos ^{3} θ - cos θ} = tan θ$ .
(viii) $(sin A + csc A)^{2} + (cos A + sec A)^{2} = 7 + tan^{2} A + cot^{2} A$ .

Answer

All eight identities are proved: in each case the LHS simplifies to the RHS, so LHS

=

RHS.

Step-by-step solution

Every step of how it is worked out, free, with no account needed.