Question

Solve
$\sf \: \sqrt{(1 - cos {}^{2} \theta) sec{}^{2} \theta } = \tan( \theta)$
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Answer 1

$\sf \sqrt{(1 \: - \: cos^2 ∅) sec^2 ∅} \: = \: tan ∅$

• Taking LHS,

$\sf \sqrt{(1 \: - \: cos^2 ∅) sec^2 ∅}$

• We know that,

sin² ∅ + cos² ∅ = 1

• Rearranging,

sin² ∅ = 1 - cos² ∅

• Substituting the values,

$\sf \sqrt{sin^2 ∅ \: * \: sec^2 ∅}$

$\sf sin ∅ \: * \: sec ∅$

• We know that,

$\sf sec ∅ \: = \: \dfrac{1}{cos ∅}$

• Substituting the values,

$\sf sin ∅ \: * \: \dfrac{1}{cos ∅}$

$\sf \dfrac{sin ∅}{cos ∅}$

• We know that,

$\sf \dfrac{sin ∅}{cos ∅} \: = \: tan ∅$

$\sf = \: tan ∅$

• LHS = RHS

• Hence, proved.

Answer 2

$\sf\underline\pink{Question}$

To Proof:-

• $\sf \sqrt{(1 \: - \: cos^2 ∅) sec^2 ∅} \: = \: tan ∅$

$\sf\underline\purple{step-by-step\:solution}$

Using Formula:

• (1) sin² ∅ + cos² ∅ = 1

• (2) $\sf sec ∅ \: = \: \dfrac{1}{cos ∅}$

• (3) $\sf \dfrac{sin ∅}{cos ∅} \: = \: tan ∅$

Taking LHS,

$\implies \sf \sqrt{(1 \: - \: cos^2 ∅) sec^2 ∅}$

Rearranging formula in terms of Sin ∅

sin² ∅ = 1 - cos² ∅

Substituting the values in formula (1):

$\implies \sf \sqrt{sin^2 ∅ \: * \: sec^2 ∅}$

$\implies\sf sin ∅ \: * \: sec ∅$

Substituting the values in formula (2):

$\implies\sf sin ∅ \: * \: \dfrac{1}{cos ∅}$

$\implies\sf \dfrac{sin ∅}{cos ∅}$

By formula (3)

$\sf = \: tan ∅$

L.H.S = R.H.S

Hence, proved!!