Question

evaluate 2tan45°÷1+tan^2 45°​

Answer 1

Step-by-step explanation:

$\leadsto$ Method 1:-

$\frac{2 \tan(45°) }{1+ \tan^2(45) }$

We know that,

sin2A = $\frac{2 \tan(45°) }{1+ \tan^2(45) }$

Therefore,

sin2(45°) = $\frac{2 \tan(45°) }{1+ \tan^2(45) }$

sin90° = $\frac{2 \tan(45°) }{1+ \tan^2(45) }$

We know that sin90° = 1,

Therefore,

$\frac{2 \tan(45°) }{1+ \tan^2(45) }$ = 1

$\leadsto$ Method 2:-

We know that tan 45° = 1

Therefore,

$\frac{2 \tan(45°) }{1+ \tan^2(45) }$ = 2/(1 + 1) = 2/2 = 1

$\leadsto$ Derivation of sin2A in terms of tanA

We know that sin2A = 2sinAcosA

Multiply and divide by cosA in sin2A

sin2A = 2sinAcosA * cosA / cosA = 2tanAcos²A

We know that cosA = 1/secA and secA = 1/cosA

Therefore,

sin2A = 2tanA/sec²A

We know that,

sec²A = 1 + tan²A

Therefore,

sin2A = 2tanA/(1 + tan²A)