JIPMAT 2022 (QA) - Which of the following trigonometric identities are true? aligned & ^2(41^)+ ^2(49^)=1 \\ & ^2(60^)-2 (45^)- ^2(30^)=-1 \\ & ^2()+1/1+ ^2()=1 aligned | PYQs + Solutions

JIPMAT 2022

Geometry

Trigonometry

Medium

Which of the following trigonometric identities are true?
$\begin{aligned} & \sin ^2\left(41^{\circ}\right)+\sin ^2\left(49^{\circ}\right)=1 \\ & \sin ^2\left(60^{\circ}\right)-2 \tan \left(45^{\circ}\right)-\cos ^2\left(30^{\circ}\right)=-1 \\ & \sin ^2(\theta)+\frac{1}{1+\tan ^2(\theta)}=1 \end{aligned}$

A and B only

A and C only

B and C only

A, B, and C

Correct Option: 2

For statement A:

\sin^2(41°) + \sin^2(49°) = \sin^2(41°) + \sin^2(90° - 41°)

= \sin^2(41°) + \cos^2(41°) = 1

✓ [We know that

\sin^2(x)+\cos^2(x)=1

]

For statement B:

\sin^2(60°) - 2\tan(45°) - \cos^2(30°)

= \frac{3}{4} - 2(1) - \frac{3}{4} = -2 ≠ -1

✗

For statement C:

\sin^2(\theta) + \frac{1}{1 + \tan^2(\theta)}

= \sin^2(\theta) + \cos^2(\theta)

(since

\frac{1}{1 + \tan^2(\theta)} = \cos^2(\theta)

)

= 1

✓

Hence, A and C are true.