Prove the identity. (sec^(2)x-1)cos^(2)x=sin^(2)x

Apply trigonometric identity: We start with the left-hand side of the equation (sec^(2)x-1)cos^(2)x and apply the trigonometric identity sec^2(x) = 1 + tan^2(x). (sec^(2)x-1)cos^(2)x = (1 + tan^2(x) - 1)cos^2(x)Simplify expression: Simplify the expression by canceling out the 1 and -1. (1 + tan^2(x) - 1)cos^2(x) = tan^2(x)cos^2(x)Rewrite using identity: Use the trigonometric identity tan(x) = sin(x)/cos(x) to rewrite tan^2(x) as (sin^2(x)/cos^2(x)). tan^2(x)cos^2(x) = (sin^2(x)/cos^2(x))cos^2(x)Cancel out terms: Simplify the expression by canceling out the cos^2(x) terms. (sin^2(x)/cos^2(x))cos^2(x) = sin^2(x)Prove identity: We have now shown that the left-hand side of the equation simplifies to sin^2(x), which is the right-hand side of the original equation. Therefore, the identity is proven. sin^2(x) = sin^2(x)

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Prove the identity.
(sec^(2)x-1)cos^(2)x=sin^(2)x

Prove the identity. $\newline$ $\left(\sec ^{2} x-1\right) \cos ^{2} x=\sin ^{2} x$

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Grade 7

One-step inequalities: word problems

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Q. Prove the identity.

\newline

\left(\sec ^{2} x-1\right) \cos ^{2} x=\sin ^{2} x

Apply trigonometric identity: We start with the left-hand side of the equation $(\sec^{2}x-1)\cos^{2}x$ and apply the trigonometric identity $\sec^2(x) = 1 + \tan^2(x)$ . $\newline$ $(\sec^{2}x-1)\cos^{2}x = (1 + \tan^2(x) - 1)\cos^2(x)$
Simplify expression: Simplify the expression by canceling out the $1$ and $-1$ . $(1 + \tan^2(x) - 1)\cos^2(x) = \tan^2(x)\cos^2(x)$
Rewrite using identity: Use the trigonometric identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$ to rewrite $\tan^2(x)$ as $\left(\frac{\sin^2(x)}{\cos^2(x)}\right)$ . $\newline$ \tan^\(2(x)\cos^ $2$ (x) = \left(\frac{\sin^ $2$ (x)}{\cos^ $2$ (x)}\right)\cos^ $2$ (x)
Cancel out terms: Simplify the expression by canceling out the $\cos^2(x)$ terms. $\frac{\sin^2(x)}{\cos^2(x)}\cos^2(x) = \sin^2(x)$
Prove identity: We have now shown that the left-hand side of the equation simplifies to $\sin^2(x)$ , which is the right-hand side of the original equation. Therefore, the identity is proven. $\newline$ $\sin^2(x) = \sin^2(x)$