We have to prove, tan^2A sin^2A=sin^4A^2A tan^2A sin^2A ⇒ #160; sin^2Acos^2A sin^2A ⇒ #160; sin^2A sin^2A.cos^2Acos^2A ...

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Trigonometric Identities

tan2A-sin2A=s...

Question

${tan}^{2} A - {sin}^{2} A = {sin}^{4} A {sec}^{2} A$ .

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Solution

We have to prove, ${tan}^{2} A - {sin}^{2} A = {sin}^{4} A {sec}^{2} A$

${tan}^{2} A - {sin}^{2} A$

$\Rightarrow$ $\frac{{sin}^{2} A}{{cos}^{2} A} - {sin}^{2} A$

$\Rightarrow$ $\frac{{sin}^{2} A - {sin}^{2} A . {cos}^{2} A}{{cos}^{2} A}$

$\Rightarrow$ $\frac{{sin}^{2} A (1 - {cos}^{2} A)}{{cos}^{2} A}$

$\Rightarrow$ $\frac{{sin}^{2} A . {sin}^{2} A}{{cos}^{2} A}$

$\Rightarrow$ $\frac{{sin}^{4} A}{{cos}^{2} A}$

$\Rightarrow$ ${sin}^{4} A . \frac{1}{{cos}^{2} A}$

$\Rightarrow$ ${sin}^{4} A . {sec}^{2} A$

$∴$ ${tan}^{2} A - {sin}^{2} A = {sin}^{4} A {sec}^{2} A$ [ Hence proved ]

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Similar questions

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{tan}^{2} A - {sin}^{2} A = {sin}^{4} A {sec}^{2} A

\frac{{cos}^{2} A + {tan}^{2} A - 1}{{sin}^{2} A} = {tan}^{2} A .

How many of below given statements are correct?

1. sin2A = 2sinA.cosA

2.cos2A = $s i n^{2} A - c o s^{2} A$

3.tan A = $\frac{2 t a n \frac{A}{2}}{1 - t a n^{2} \frac{A}{2}}$

4. $s i n^{2} \frac{A}{2}$ = 1 - cosA

5. $t a n^{2} A$ = $\frac{1 - c o s 2 A}{1 + c o s 2 A}$

6. $s i n^{2} A$ = $\frac{2 t a n A}{1 - t a n^{2} A}$

tan²A-sin²A=tan²AsinA²

If $3 c o t A = 4 . Prove that \frac{1 - t a n^{2} A}{1 + t a n^{2} A} = c o s^{2} A - s i n^{2} A$ .

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