Question

Tan theta - cot theta├╖sin theta x cos theta = sec^2 theta- cosec^2theta
Prove that

Answer 1

GIVEN :

The trignometric equation $\frac{tan\theta-cot\theta}{sin\theta cos\theta}=sec^2\theta-cosec^2\theta$

TO PROVE :

The given trignometric equation is true.

SOLUTION :

Given trignometric equation is $\frac{tan\theta-cot\theta}{sin\theta cos\theta}=sec^2\theta-cosec^2\theta$

To prove that the given trignometric equation is true.

That is to prove that LHS=RHS

Now taking LHS $\frac{tan\theta-cot\theta}{sin\theta cos\theta}$

$\frac{tan\theta-cot\theta}{sin\theta cos\theta}$

$=\frac{tan\theta}{sin\theta cos\theta}-\frac{cot\theta}{sin\theta cos\theta}$

By using the trignometric formula:

$tanx=\frac{sinx}{cosx}$

$=\frac{\frac{sin\theta}{cos\theta}}{sin\theta cos\theta}-\frac{\frac{cos\theta}{sin\theta}}{sin\theta cos\theta}$

$=\frac{sin\theta}{cos\theta(sin\theta cos\theta)}-\frac{cos\theta}{sin\theta(sin\theta cos\theta)}$

$=\frac{1}{cos^2\theta}-\frac{1}{sin^2\theta}$

By using the trignometric formulae :

i) $\frac{1}{sinx}=cosecx$

ii) $\frac{1}{cosx}=secx$

$=sec^2\theta-cosec^2\theta$ =RHS

LHS = RHS

∴ $\frac{tan\theta-cot\theta}{sin\theta cos\theta}=sec^2\theta-cosec^2\theta$

∴ the given trignometric equation is true.

Hence proved.