Question

The value of (sin 2 20 degree + sin 2 70 degree - tan 2 45 degree) is

Answer 1

Ans is zero
Sin2 20+sin2 70=sin2 20+cos2 20=1
Tan2 45=1
1-1=0

Answer 2

The value of $\left(\sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}-\tan ^{2} 45^{\circ}\right)$ is 0  

To find the value of $\sin ^{2} 20^{\circ}+\sin ^{2}(70)^{\circ}-\tan ^{2} 45^{\circ} \rightarrow(1)$

Finding the value of each term and substituting in equation (1)

$\sin ^{2} 70^{\circ}$

By using the formula of allied angles  

$\sin (90-\theta)=\cos \theta$

Find the number which gives 70 while subtracting with 90

$\sin ^{2} 70^{\circ}=\sin ^{2}(90-20)^{\circ}$

Applying the formula $\sin ^{2}(90-20)^{\circ}=\cos ^{2} 20^{\circ} \rightarrow(2)\tan ^{2} 45^{\circ}$

From the exact values of trigonometric angles the value of $\tan 45^{\circ}=1$

Therefore, the value of $\tan ^{2} 45^{\circ}=\left(1^{2}\right)=1 \rightarrow(3)$

Substituting equation (2) and (3) in equation (1)

$\begin{array}{l}{\sin ^{2} 20^{\circ}+\sin ^{2}(70)^{\circ}-\tan ^{2} 45^{\circ}} \\ {=\sin ^{2} 20^{\circ}+\cos ^{2} 20^{\circ}-1 \rightarrow(4)}\end{array}$

From the trigonometric identities $\sin ^{2} \theta+\cos ^{2} \theta=1$

Therefore $\sin ^{2} 20^{\circ}+\cos ^{2} 20^{\circ}=1$ using the above identity

Substituting in equation (4)  

= 1-1

=0.